IN  MEMORIAM 

FLORIAN  CAJORI 


-,-',     A 


rf     \ <x.»Vi 

INTERMEDIATE 


ARITHMETIC 


ON    THE    INDUCTIVE    METHOD,    WITH    PARALLEL    MENTAL 

AND    WRITTEN    EXERCISES 


BY 

J.    W.  NICHOLSON,  A.M. 

Professor  of  Mathematics  in  the  Louisiana  State  University  and  Agricultural 

and  Mechanical  College 


NEW   ORLEANS 
PUBLISHED  BY  F.  F.  HANSELL  &  BRO. 


PRACTICAL  EDUCATIONAL  SERIES. 


Chambers'  Twenty  Lessons  in  Book-keeping. 
Hanse/l's  Penmanship. 
Nicholson's  Primary  Arithmetic. 
Nicholson's  Intermediate  Arithmetic. 
Nicholson's  Complete  Arithmetic. 


Entered  according  to  Act  of  Congress  in  the  year  1885,  by 

F.  F.  HANSELL  &  BRO., 
In  the  office  of  the  Librarian  of  Congress,  Washington,  D.  C. 


PREFACE. 


THE  chief  difference  between  a  good  and  an  inferior  Arith- 
metic is  not  so  much  a  question  of  matter  and  rules,  as  it  is  of 
method  in  the  presentation  and  development  of  principles.  In 
the  former,  few  mathematicians  would  be  bold  enough  to  lay 
claim  to  originality ;  but  in  the  latter  every  one  will,  perhaps, 
admit  there  is  room  for  improvement. 

In  the  preparation  of  this  three-book  series,  consisting  of  a 
Primary,  an  Intermediate,  and  a  Complete  Arithmetic,  the  au- 
thor has  been  influenced  by  the  following  considerations : 

1°.  Arithmetic  treats  of  the  whole  and  its  parts.  These  are 
the  magnitudes  or  objects  about  which  Analysis  and  Synthesis 
are  conversant,  and  on  the  consideration  of  which  depends  the 
solution  of  every  problem.  Hence,  the  early  introduction  of 
these  terms,  and  frequent  reference  to  them  in  the  deduction  of 
succeeding  principles,  are  of  the  greatest  importance. 

2°.  By  Induction  a  pupil  is  led  by  easy  steps,  by  familiar 
illustrations  and  commonplace  parallelisms,  into  a  clear  appre- 
hension of  principles  and  definitions.  Hence,  each  subject 
should  be  introduced  with  inductive  exercises. 

3°.  Pupils  advance  intelligently  in  any  new  subject  just  in 
proportion  as  they  perceive  in  it  a  continuation  of  the  princi- 
ples with  which  they  are  familiar.  Hence,  whatever  of  same- 
ness and  of  difference  there  is  in  the  old  and  the  new  should 
be  made  as  conspicuous  as  possible. 

4°.  Mental  and  written  work  are  equally  important,  and  should 
be  mutually  supplemental.  A  problem  intended  for  written 
work  should,  in  general,  be  preceded  by  a  parallel  question  de- 
signed for  mental,  and  also  as  an  inductive  exercise. 

5°.  The  representing  of  objects  by  the  first  letters  of  their 
names,  as,  a  for  apple,  and  "b  for  boy  or  box,  is  not  only  a 
matter  of  convenience,  but  serves  to  lead  pupils  into  the  habit 

of  generalization. 

(iii) 


iv  Preface. 

6°.  Pictures  assist  the  child  to  some  extent  in  the  conception 
of  combining  and  resolving  numbers  by  counting,  adding,  sub- 
tracting, etc.,  but  are  not  so  useful  in  this  particular  as  objects 
themselves.  Hence,  the  introduction  of  object  exercises  is  a 
prominent  feature  of  the  first  two  books  of  the  series. 

On  the  whole,  the  series  is  not  the  product  of  preconceived 
opinions  as  to  what  should  constitute  matter  and  method,  but 
the  embodiment  of  the  results  of  many  years  experience  in 
teaching  mathematics. 

The  present  treatise  is  intended  primarily  to  prepare  pupils 
for  the  Complete  Arithmetic ;  secondly,  to  meet  the  wants  of 
those  who  desire  only  a  knowledge  of  those  practical  applica- 
tions of  numbers  which  are  most  frequently  used  in  ordinary 
business  transactions.  It  is  divided  into  two  parts. 

The  First  Part  is  devoted  to  a  few  lessons  in  primary  arith- 
metic, embodying  the  more  important  features  of  the  "  Grube 
Method,"  with  such  additions  as  to  bring  it  into  conformity 
with  the  principles  already  enunciated.  This  part  may  be 
omitted  by  those  who  have  completed  the  Primary  Arithmetic, 
at  the  discretion  of  the  teacher. 

The  Second  Part  embraces  a  very  thorough  elementary  course 
in  Notation  and  Numeration,  Addition,  Subtraction,  Multiplica- 
tion, Division,  Divisors  and  Multiples,  Common  and  Decimal 
Fractions,  United  States  Money,  Compound  Numbers,  some  Im- 
portant Practical  Applications,  and  Percentage,  including  Com- 
mission, Profit  and  Loss  and  Interest.  Special  attention  is  in- 
vited to  the  simple,  progressive,  and  practical  treatment  of  these 
subjects,  especially  Division,  Fractions,  Decimals  and  Interest. 

The  Author  acknowledges  his  indebtedness  to  many  writers 
upon  this  subject,  both  of  this  and  of  other  countries,  whose 
able  treatises  have  been  consulted  with  pleasure  and  profit. 

J.  W.  N. 

BATON  ROUGE,  LA.,  June,  1885. 


CONTENTS. 


PART  I.     INDUCTIVE  ARITHMETIC. 

PAGE 

LESSON    I.     Counting  10 7 

II.    The  Number  Two 9 

III.  The  Number  Three 11 

IV.  The  Number  Four 14 

V.    The  Number  Five 17 

"      VI.    The  Number  Six 21 

"    VII.    The  Number  Seven 25 

"  VIII.     The  Number  Eight 28 

"      IX.     The  Number  Nine 31 

"        X.     The  Number  Ten .  35 


PART   II. 

Notation  and  Numeration 41 

Addition 54 

Subtraction 71 

Multiplication 87 

Division 102 

Short  Division 113 

Long  Division 115 

Divisors  and  Multiples 126 

Common  Fractions 132 

Reduction  138 

Addition 146 

Subtraction 148 

Multiplication 152 

Division 156 

Decimal  Fractions  164 

Reduction 168 

Addition 170 


VI 


Contents. 


Decimal  Fractions—  PAGE 

Subtraction 171 

Multiplication 172 

Division 172 

United  States  Money  174 

Reduction 175 

Addition,  Subtraction,  Multiplication,  and  Division 177 

Bills  and  Accounts 178 

Compound  Numbers 181 

Linear  Measure 181 

Reduction 182 

Square  Measure 183 

Solid  or  Cubic  Measure 185 

Liquid  Measure 186 

Dry  Measure 188 

Troy  Weight 189 

Avoirdupois  Weight 190 

Apothecaries  Weight 191 

United  States  Money 192 

English  Money „  .  192 

French  Money ...  193 

Measure  of  Time 193 

Circular  Measure 195 

Paper  Measure  and  Miscellaneous  Table 196 

The  Old  French  Measure 196 

Compound  Addition.  198 

Compound  Subtraction 198 

Compound  Multiplication 200 

Compound  Division 200 

Important  Applications 204 

To  find  the  time  between  two  dates 204 

To  find  the  area  of  rectangular  surfaces 205 

To  find  the  volume  of  rectangular  solids 206 

To  find  the  capacity  of  tanks  in  gallons 207 

To  find  the  capacity  of  bins  or  granaries  in  bushels.  . .  208 
To  find  the  number  of  board  feet  in  a  plank  or  board.  208 

Percentage 211 

Commission 214 

Profit  and  Loss 215 

Interest.  217 


PART  I. 


INTERMEDIATE  ARITHMETIC. 


LESSON  I. 

COUNTING  TEN. 

First  Row  ...     a 

Second    "  ...     a     a 

Third     "  ...    a    a    a 

Fourth  "  ...    a    a    a    a 

Fifth      "  ...    a    a    a    a    a 

Sixth     "  ...    a    a    a    a    a    a 

Seventh "  ...aaaaaaa 

Eighth  "  ...aaaaaaaa 

Ninth     "  ...    a    a    a    a    a    a    a    a    a 

Tenth    "  ...aaaaaaaaaa 

DIAGRAM  OF  A's. 

Count  the  a's  in  each  row,  beginning  at  the  top. 

Count  the  a's  in  each  row,  beginning  at  the  bottom. 

Which  is  the  first  row?  The  second?  The  third? 
The  fourth  ?  etc. 

How  many  a's  are  in  the  first  row?  The  third  row? 
The  ninth?  The  fifth?  The  second?  The  seventh? 
The  fourth?  The  eighth?  The  sixth?  The  tenth? 

(7) 


8  Intermediate  Arithmetic. 


Figure  0  stands  for  none, 
"      1       "        "    one, 
"      2       "        "    two, 
"      3       "        "    three, 
"      4      "        "    four, 


Figure  5  stands  for  five, 


<.        a        a          u 


6  "    six, 


a          7          " 


7  "    seven, 


8  "    eight, 


9  "    nine, 


10  stands  for  ten. 

Which  row  has  two  a's  in  it?    5  a's?    9  a's?     7  a's? 

Make  6  taps  on  your  slate.  Make  three  marks.  Get 
up  and  make  8  steps.  Hold  up  10  fingers;  5  fingers;  3 
fingers. 

Make  all  the  figures  on  your  slate.     Thus : 

01234-56    7    89    10 

Count  10  axes.  Ans.  1  ax,  2  axes,  3  axes,  4  axes, 
5  axes,  6  axes,  7  axes,  8  axes,  9  axes,  10  axes. 

Count  10  arms,  10  apples. 

What  may  a  stand  for?  Ans.  An  ax,  or  an  arm,  or 
an  apple. 

Count  10  a's.  Ans.  1  a,  2  a,  3  a,  4  a,  5  a,  6  a,  7  a, 
8  a,  9  a,  10  a. 

Count  8  boys ;  6  bats ;  7  birds. 

What  stands  for  a  boy,  or  a  bat,  or  a  bird,  etc.? 
Ans.  b. 

Count  9  b's. 

What  stands  for  a  cat,  or  a  cow,  or  a  cap,  etc.?    Ans.  c. 

Count  5  c's. 

What  stands  for  a  horse,  or  an  hour,  or  a  hen? 

What  stands  for  a  finger,  or  a  fan,  or  a  fob? 

What  is  a  Unit?     Ans.  One  of  the  things  counted. 
What  is  the  unit  of  8  fans?    9  boys?    7  girls?    5  a's? 
Ans.  1  a. 


The  Number  Two. 


SUGGESTIONS  TO  TEACHERS.  —I.  This  first  lesson  should  be  re- 
cited with  each  of  the  following  until  it  is  well  learned.  Copy 
the  diagram  of  ct's  on  the  board  for  recitation,  occasionally  sub- 
stituting other  letters  for  a.  In  this,  as  in  subsequent  lessons, 
ask  such  additional  questions  as  will  enlist  the  interest  and  meet 
the  wants  of  the  individual  pupil. 

II.  It  is  important  that  the  pupil  should  obtain  a  clear  idea 
of  the  number  of  units  represented  by  a  figure.     Hence,  it  is 
recommended  that  the  teacher  write  on  the  board  some  figure, 
as  7,  without  calling  its  name,  and  require  the  pupil  to  make 
that  many  marks,  or  steps,  or  hold  up  that  many  fingers     Pu- 
pils should  practice  writing   the   figures  until   they  can  make 
them  readily  and  correctly. 

III.  The  sign  -f  should  always  be  read  and,  and  —  less,  and 
not  plus  and  minus,  until  the  student  becomes  perfectly  familiar 
with  their  primary  meaning. 

IV.  In  illustrating  principles  and  processes,  use  material  ob- 
jects as  far  as  possible. 

NOTE. --Prior  to  the  subject  of  Fractions,  the  term  parts  de- 
note integers,  and  the  two  complemented  parts  of  a  number  are 
denoted  by  the  two  parts. 


LESSON  II. 

ABOUT  THE  NUMBER  Two. 


What  lesson  is  this? 
What  is  it  about? 
Hold  up  two  fingers. 
How  many  hands 
have  you?  Write  2 
on  your  slate.  Write 
2  two  times.  How 
many  horses  are  in 
the  picture?  Do  2 
horses  make  a  pair? 


^-syt.lfiaU      *~1--- •   IrrfHjr^^r-f,:-" • .."" -- : 


10  Intermediate  Arithmetic. 

Is  1  horse  the  whole  or  a  part  of  the  pair?  What  are 
the  two  parts  of  2  horses?  Ans.  1  horse  and  1  horse. 
What  are  the  two  parts  of  2  men?  2  cows?  2  dollars? 
2  houses?  2  a's?  2  b's? 

What  are  the  two  parts  of  2?     Ans.  1  and  1. 

ADDITION.  — Uniting  the  parts. 

How  many  are  1  horse  and  1  horse?  1  cow  and  1 
cow?  1  mule  and  1  mule?  1  apple  and  1  apple?  1 
b  and  1  b?  1  and  1? 

What  sign  stands  for  and?  Ans.  +.  What  is  it 
called?  Ans.  And.  Go  to  the  board  and  make  it. 
What  sign  stands  for  are  and  equals?  Ans.  =.  Make 
it  on  the  board.  Write  this:  1  +  1=2  on  the  board. 
How  is  it  read?  Ans.  1  and  1  are  2. 

SUBTRACTION.  —  Taking  away  one  part. 

Two  horses  are  together;  if  1  horse  is  taken  away, 
how  many  horses  will  be  left?  1  horse  taken  from  2 
horses  leaves  how  many?  Jane  had  2  roses,  but  gave 

1  rose  to  Mary;    how  many  roses  did  Jane  have  left? 
How  many  are  2  roses  less   1  rose?     2  peaches  less   1 
peach?     2  pins  less  1  pin?     2  birds  less  1  bird? 

What  sign  stands  for  less?  Ans.  — .  Make  it  on  the 
board.  What  is  it  called?  Ans.  Less.  Write  this: 

2  — 1  =  1  on  the  board.     How  is  it  read?     Ans.  2  less 
1  is  1,  or  1  from  2  leaves  1. 

MULTIPLICATION,  —Uniting  equal  parts. 

Make  1  mark.  Make  1  mark  2  times.  How  many 
are  2  times  1  mark?  2  times  1  horse?  Hold  up  2 
fingers.  Now  hold  up  2  times  1  finger.  Frank  has  2 


The  Number  Three. 


11 


dimes,  and  Charles  has  2  times  1  dime ;  which  has  the 
more  money? 

What  sign  stands  for  times?  Ans.  X.  Make  it  on 
the  board.  Write  this:  2x1  =  2  on  the  board.  How 
is  it  read?  Ans.  2  times  1  are  2. 

DIVISION.  —  Measuring  by  a  part. 

Does  2  contain  its  parts?  What  are  its  parts?  Does 
2  contain  1  and  1?  Does  it  contain  1  2  times? 

How  many  times  do  2  horses  contain  1  horse?  How 
many  times  can  you  take  1  pint  from  2  pints?  How 
many  times,  then,  do  2  pints  contain  1  pint? 

What  sign  stands  for  contains?  Ans.  -K  Make  it  on 
the  board.  Write  this  :  2  -f- 1  =  2  on  the  board.  How 
is  it  read?  Ans.  2  contains  1,2  times,  or  2  divided  by 
1  equals  2. 


LESSON  III. 

ABOUT  THE  NUMBER  THREE. 

What  lesson  is  this  ? 
What  is  it  about? 
Hold  up  3  fingers. 
Tap  your  slate  3 
times.  Make  3  on 
the  board.  Make  3 
three  times.  Is  the 
wagon  loaded  with  3 
bales?  Are  3  bales 
the  whole  load  or  a 
part  of  the  load?  Are  2  bales  the  whole  load  or  a  part 


12  Intermediate  Arithmetic. 

of  the  load  ?  Is  1  bale  a  part  of  the  load  ?  Do  2  bales 
and  1  bale  make  3  bales  ?  Are  1  bale  and  2  bales  the 
same  as  2  bales  and  1  bale?  What  are  the  two  parts 
of  3  bales?  Ans.  1  bale  and  2  bales.  What  are  the 
two  parts  of  3  barrels?  3  birds?  3  dollars?  3  <Ts? 
3  ill's? 

What  are  the  two  parts  of  3?  Ans.  1  and  2,  or  2 
and  1. 

ADDITION.  —  Uniting  the  parts. 

How  many  are  2  bales  and  1  bale  when  put  together? 
2  horses  and  1  horse?  1  dime  and  2  dimes?  2  d's  and 
Id?  1  C  and  2  c's?  1  and  2?  What  are  the  two 
parts  of  3?  When  you  put  them  together,  do  they 
make  3  ?  Ann  has  2  plums  and  Susan  has  1  plum ; 
how  many  plums  have  they  together?  Ben  has  one 
marble  and  Ike  has  2  marbles;  if  they  put  them  in  a 
sack  how  many  will  be  in  the  sack? 

Write  these,  24-1=3  and  1  -f  2  =  3,  and  read  them. 

SUBTRACTION.  —Taking  away  one  part. 

How  many  bales  would  be  left  on  the  wagon  if  1 
bale  were  rolled  off?  If  2  bales  were  rolled  off?  How 
many  are  left  when  1  is  taken  from  3?  When  2  is 
taken  from  3?  What  are  the  two  parts  of  3?  When 
one  of  them  is  taken  from  3,  is  the  other  left?  Jane 
had  3  roses,  but  gave  1  rose  to  Mary ;  how  many  roses 
did  Jane  have  left?  Ann  had  3  cherries  but  gave  2 
cherries  to  Ben  ;  how  many  did  Ann  then  have  ?  How 
many  are  3  roses  less  1  rose?  3  cherries  less  2  cher- 
ries? 3  r's  less  2  r's?  3  c's  less  1  c?  3  less  1? 

Copy  and  read,  3  —  1  =  2;  3  —  2  =  1;  3  —  3  =  0. 

James  killed  1  bird  and  John  killed  3  birds;  how 
many  more  did  John  kill  than  James? 


The  Table  of  Three.  13 

MULTIPLICATION.—  Uniting  equal  parts. 

How  many  are  1  and  1  and  1  ?  How  many  times  is 
1  taken  ?  3  times  1  are  how  many  ?  How  many  are  3 
times  1  bale?  3  times  1  horse?  3  times  1  h?  3  times 
la?  Julia  has  1  rose,  and  Mary  has  3  times  as  many 
as  Julia;  how  many  roses  has  Mary?  Show  me  3  fin- 
gers. Now  show  me  3  times  1  finger. 

Copy  and  read,  3  X  1=3. 


DIVISION.  —  Measuring  by  a  part. 

Does  the  load, — 3  bales,  contain  its  parts?  Does  it 
contain  1  bale  and  1  bale  and  1  bale?  Does  it  contain 
1  bale  3  times?  Put  3  books  on  the  table.  Now  take 
off  1  book  at  a  time  until  all  are  removed.  How  many 
times  did  you  take  off  1  book?  How  many  times, 
then,  do  3  books  contain  1  book?  If  a  boy  carries  off 
a  bushel  of  corn  at  a  time,  how  many  trips  will  he 
have  to  make  to  carry  off  3  bushels  ?  How  many  times, 
then,  do  3  bushels  contain  1  bushel? 

Copy  and  read,  3  -*-  1  =  3 ;  3  -f-  3  =  1. 

Learn  and  recite : 

THE  TABLE  OF  THREE. 

0  and      3  are  3  0  from  3  leaves  3 

1  and      2  are  3  1  from  3  leaves  2 

2  and      1  are  3  2  from  3  leaves  1 

3  and     0  are  3  3  from  3  leaves  0 
3  times  1  are  3  1  in       33  times 

NOTE. — 0  is  called  none ;  thus,  none  and  3  are  3. 


14 


Intermediate  Arithmetic. 


LESSON  IV. 

ABOUT  THE  NUMBER  FOUR. 

What  lesson  is  this?  What  is  it  about?  Is  4  more 
than  3?  How  many  more?  Show  me  4  books.  In  the 

picture  is  a  class  of 
girls  ;  how  many  girls 
are  there?  Are  3 
girls  the  whole,  or  a 
part  of  the  class? 
Are  2  girls  the  whole, 
or  a  part  of  the 
class?  Is  1  girl  a 
part  of  the  class  ?  If 
the  class  were  divided  into  two  parts,  how  many  girls 
would  be  in  each  part?  Ans.  1  girl  in  one  part  and 
3  girls  in  the  other  part ;  or  2  girls  in  one  and  2  girls 
in  the  other. 

Show  me,  with  4  fingers,  how  this  would  be.  What, 
then,  are  the  two  parts  of  4  girls  ?  Ans.  1  girl  and  3 
girls,  and  2  girls  and  2  girls. 

What  are  the  two  parts  of  4?    Ans.  1  and  3,  2  and  2. 

ADDITION.—  Uniting  the  parts. 

How  many  are  1  girl  and  3  girls?  3  girls  and  1 
girl?  2  girls  and  2  girls?  3  horses  and  1  horse?  2 
mules  and  2  mules?  1  ill  and  3  in's?  3  ii's  and  1  11? 
2  and  2? 

What  are  the  two  parts  of  4?  When  you  put  them 
together,  do  they  make  4?  Ann  has  1  plum  and 
Emma  has  3  plums  ;  how  many  plums  have  they  to- 
gether? Ben  has  2  marbles  and  Jake  has  2  marbles; 


The  Number  Four.  15 

how  many  have  they  together?  Julia  has  3  dolls  and 
Mary  has  1  doll  more  than  Julia;  how  many  dolls  has 
Mary  ? 

Copy  and  read :  1  +  3  =  4;  2  +  2  =  4;  3  +  1—4. 

SUBTRACTION.  —Taking  away  one  part. 

How  many  girls  would  be  left  in  the  class,  if  1  girl 
were  taken  away  ?  If  2  girls  were  taken  away  ?  If  3 
girls?  If  4  girls?  1  from  4  leaves  how  many?  2  from 
4  leaves  how  many?  3  from  4?  4  from  4? 

What  are  the  two  parts  of  4?  When  one  of  them 
is  taken  from  4,  is  the  other  left?  Jane  had  4  roses 
but  gave  her  sister  1  rose ;  how  many  roses  did  Jane 
then  have?  Frank  had  4  dimes  but  gave  2  dimes  to 
an  old  blind  man,  how  many  dimes  did  Frank  then 
have  ?  4  birds  were  on  a  limb,  but  3  birds  have  flown  ; 
how  many  birds  are  left? 

Copy  and  read:   4  —  1  =  3;   4  —  2  =  2;    4  —  3  =  1. 

MULTIPLICATION  .—Uniting  equal  parts. 

How  many  are  1,  1,  1  and  1?  How  many  times  is 
the  part  1  taken  ?  How  many,  then,  are  4  times  1  ? 
How  many  are  4  times  1  girl?  4  times  1  book?  4 
times  1  horse? 

How  many  are  2  and  2  ?  How  many  times  is  the 
part  2  taken?  How  many,  then,  are  two  times  2? 
How  many  are  2  times  2  girls?  2  times  2  birds?  2 
times  2  roses?  Show  me  4  fingers.  Now  show  me  4 
times  1  finger.  Now  show  me  2  times  2  fingers.  Ben 
has  4  times  1  toy  and  Frank  has  2  times  2  toys ;  which 
has  the  more? 

Copy  and  read  :  4x1  =  4;  2  X  2  =  =  4. 


16  Intermediate  Arithmetic. 

DIVISION.  — Measuring  by  a  part. 

Does   4    contain  its   parts?     Does   it  contain  1,  1,  1, 
and  1  ?     How   many   times   does   it   contain  1  ?     How 
many  times  can  1   gill  be   taken   from   4  gills?     How 
many  times,  then,  do  4  gills   contain    1   gill?     Does  4 
contain  the  parts  2  and  2?      How   many  times,  then, 
does  it  contain  2  ?     Are  4  gills  the  same  as  2  gills  and 

2  gills?     How  many  times,  then,  do  4  gills  contain  2 
gills?     How  many  boxes  shall  I  need   so   that  I  may 
put  4    marbles  in    them  and   have   2    marbles  in   each 
box? 

Copy  and  read:  4-=-l=4;  4-^2  =  2;  4-M  =  l. 

OBJECT  EXERCISES. 

Take  4  blocks,  put  them  on  a  table,  and  call  them 
birds.  Now  divide  or  separate  the  birds  into  two  parts, 
every  way  you  can,  calling  the  results  thus :  1  bird  and 

3  birds  are  4  birds,  etc. 

Now  take  off  a  bird  at  a  time,  calling  the  results 
thus :  1  bird  from  4  birds  leaves  3  birds  :  2  birds  from 

4  birds  leave   2    birds,  etc.      Now   put   the  birds  back, 
one  at  a  time,  calling  the  results  thus :  1  time  1  bird  is 
1  bird ;  2  times  1  bird  are  2  birds,  etc.     Next,  take  the 
birds  off,  one  at  a  time,  calling  the  results  thus :  1  bird 
contains  1  bird  1  time;  2  birds  contain  1  bird  2  times, 
etc. 

Next,  put  the  birds  back,  two  at  a  time,  calling  the 
results  thus :  1  time  2  birds  is  2  birds ;  2  times  2  birds 
are  4  birds. 

Now  take  the  birds  off,  two  at  a  time,  calling  tho 
results  thus :  2  birds  contain  2  birds  1  time ;  4  birds 
contain  2  birds  2  times. 


The  Number  Flee. 


THE  TABLE  OF  FOUR. 


0  and  4  are  4 

1  and  3  are  4 

2  and  2  are  4 

3  and  1  are  4 

4  times  1  are  4 
2  times  2  are  4 


4  from  4  leaves  0 

3  from  4  leaves  1 

2  from  4  leaves  2 

1  from  4  leaves  3 

1  in       4,  4 -times 

2  in       4,  2  times 


LESSON  V. 

ABOUT  THE  NUMBER  FIVE. 

What  lesson  is  this?  What  is  it  about?  Have  you 
5  fingers  on  one  hand?  Can  you  take  5  blocks  and 
separate  them  as  they  are 
here  pictured  ?  Take  5  rocks 
and  separate  them  as  the 
blocks  are.  Which  are  the 
first  two  parts  of  5  blocks? 
The  second?  The  third? 
The  fourth?  Are  2  blocks 
and  3  blocks  the  same  as  3 
blocks  and  2  blocks?  Are 





1  block  and  4  blocks  the  same  as  4  blocks  and  1  block? 
What  are  the  two  parts  of  5  ?     Ans.  1  and  4,  2  and  3. 

A  D  D  I  T I  0  N  .  - -Un itiny  tJic  parts. 

Count  5.  How  many  are  1  block  and  4  blocks?  2 
horses  and  3  horses?  3  men  and  2  men?  1  cow  and 
4  cows  ? 

N.  I.— 2. 


18  Intermediate  Arithmetic. 

What  are  the  two  parts  of  5  ?  Do  they  make  5  when 
united?  How  many  is  1  more  than  4?  Two  more 
than  3? 

Some  boys  are  in  a  class;  if  3  boys  are  a  part  of  the 
class,  and  2  boys  are  the  other  part,  how  many  boys 
are  in  the  class?  If  4  mules  and  1  mule  are  the  two 
parts  of  a -team,  how  many  are  in  the  team?  How 
many  are  3  girls  and  2  girls?  4  girls  and  1  girl?  2 
apples  and  3  apples?  1  a  and  4  a's? 

Copy  and  read:  4  +  1  —  5;  3  +  2  =  5;  2  +  3  =  5; 
1  +  4  =  5. 

SUBTRACTION.  —  Taking  away  one  part. 

Count  5  backward.  5  blocks  are  together ;  if  1  block 
is  removed,  how  many  blocks  will  be  left?  How  many 
are  1  block  from  5  blocks?  2  horses  from  5  horses? 
3  men  from  5  men  ?  4  cows  from  5  cows  ? 

What  are  the  two  parts  of  5  ?  When  one  of  them  is 
taken  from  5,  is  the  other  left?  How  many  are  1  less 
than  5  ?  2  less  than  5  ?  3  less  than  5  ?  etc. 

If  3  chicks  are  one  part  of  a  brood  of  5  chicks,  how 
many  chicks  are  in  the  other  part?  2  cats  are  a  part 
of  5  cats;  what  is  the  other  part? 

Copy  and  read :  5  —  1  =  4;  5  —  2  =  3;  5  —  3  =  2; 
5  —  4  =  1;  5—5  =  0. 

MULTIPLICATION.  —Uniting  equal  parts. 

How  many  are  1,  1,  1,  1,  and  1  ?  How  many  times 
is  the  part  1  taken  ?  How  many,  then,  are  5  times  1  ? 
How  many  are  5  times  1  block?  Hold  up  5  fingers. 
Now  hold  up  5  times  1  finger.  Thomas  has  5  dollars, 
and  Henry  has  5  times  1  dollar;  which  boy  has  the 


Object  Exercises.  19 

more?  Are  2  cents  and  3  cents  the  same  as  5  times 
1  cent? 

What  does  0  stand  for?  Are  5  nothings  any  more 
than  1  nothing?  How  many,  then,  are  5  times  0? 

Copy  and  read:  5  X  1:  =5  ;  5  X0  =  =  0. 

DIVISION.—  Measuring  by  a  part. 

Does  5  contain  its  parts?  Is  5  formed  of  5  ones? 
How  many  times,  then,  does  it  contain  1  ?  Make  5 
a's.  Now  erase  1  a  at  a  time  until  all  are  gone;  how 
many  times  did  you  erase?  How  many  times,  then, 
do  5  a's  contain  la? 

Five  gallons  of  water  are  in  a  tub ;  if  each  horse 
drinks  a  gallon,  how  many  horses  will  drink  it  all? 
How  many  times,  then,  is  1  gallon  contained  in  5 
gallons  ? 

Copy  and  read  :  5  -f- 1  =  5  ;  5  -=-  5  =  =  1. 

OBJECT  EXERCISES. 

Take  5  blocks,  put  them  on  a  table,  and  call  them 
hats.  Now  separate  the  hats  into  two  parts  every  way 
you  can,  calling  the  results  thus :  1  hat  and  4  hats  are 
5  hats,  etc. 

Now  take  off  1  hat  at  a  time,  calling  the  results 
thus :  1  hat  from  5  hats  leaves  4  hats ;  2  hats  from 
5  hats  leave  3  hats,  etc. 

Now  put  back  1  hat  at  a  time,  calling  the  results 
thus :  1  time  1  hat  is  1  hat ;  2  times  1  hat  are  2 
hats,  etc. 

Next  take  off  1  hat  at  a  time,  calling  the  results 
thus :  1  hat  contains  1  hat  1  time ;  2  hats  contain  1 
hat  2  times,  etc. 


20  Intermediate  Arithmetic. 

THE  TABLE  OF  FIVE. 

0  and  5  are  5  0  from  5  leaves  5 

1  and  4  are  5  1  from  5  leaves  4 

2  and  3  are  5  2  from  5  leaves  3 

3  and  2  are  5  3  from  5  leaves  2 

4  and  1  are  5  4  from  5  leaves  1 

5  and  0  are  5  5  from  5  leaves  0 

5  times  1  are  5  1  in       5,  5  times 

REVIEW  QUESTIONS. 

What  are  the  two  parts  of  2?    3?    4?    5? 

How  many  are  1  girl  and  1  girl  ?     1  and  1  ? 

How  many  are  2  boys  and  1  boy  ?     1  boy  and  2  boys  ? 

How  many  are  1  peach  and  3  peaches?    2  flies  and  2  flies? 

How  many  are  3  men  and  2  men  ?     1  mule  and  4  mules  ? 

How  many  are  left  when  1  girl  is  taken  from  2  girls? 
When  2  men  are  taken  from  3  men?  When  1  ill  is  taken 
from  3  ni's?  When  1  bird  is  taken  from  4  birds?  When  2 
b's  are  taken  from  4  b's?  When  3  pins  are  taken  from  4 
pins?  When  1  book  is  taken  from  5  books?  When  2  fingers 
are  taken  from  5  fingers?  When  3  f's  are  taken  from  5  f's? 
When  4  guns  are  taken  from  5  guns? 

How  many  are  2  times  1  gun?  2  times  2  guns?  4  times  1 
horse?  3  times  1  mule?  5  times  1  cow?  How  many  does  2 
lack  of  being  3  times  1  ?  Does  3  lack  of  being  4  times  1  ? 
Does  3  lack  of  being  2  times  2  ?  Does  3  lack  of  being  5  times  1  ? 

Plow  many  times  do  2  cents  contain  1  cent?  Do  3  bushels 
contain  1  bushel  ?  Do  4  books  contain  2  books  ?  Do  5  dollars 
contain  1  dollar?  Do  4  pins  contain  1  pin  5  times? 

James  has  1  marble  and  John  has  3  marbles.  How  many 
marbles  have  James  and  John  together?  How  many  more 
marbles  has  John  than  James?  How  many  does  John  lack  of 
having  4  times  as  many  as  James? 

Susan   has  2   roses  and   Mary  has  3  roses;   how  many  roses 


The  Number  Six. 


21 


have  both  girls?  How  many  more  roses  lias  Mary  than  Susan? 
How  many  roses  does  Mary  lack  of  having  "2  times  as  many  as 
Susan?  If  Mary  should  give  Susan  2  of  her  roses,  how  many 
would  each  girl  then  have? 

How  many  are  1  +  1  ?  2  +  2?  3  +  1?  4+1?  1+3?  3  +  2? 
4  +  0?  2  +  3?  2  —  1?  3—2?  5  —  3?  5  —  2?  4  —  3?  3—1? 
4  —  2?  5  —  4?  4  —  4?  5X0?  5X1?  2X2?  4^-1?  4-^2? 


NOTE.  --The  teacher  should  improvise  questions,  similar  to 
foregoing,  after  each  lesson.  No  question,  however,  should  in- 
volve a  number  greater  than  the  subject  of  the  last  lesson.  The 
preceding  questions  may  be  used  with  different  numbers. 


LESSON  VI. 


ABOUT  THE  NUMBER  Six. 

What  lesson  is  this?  What  is  it  about?  Show  me 
6  planks.  How  many  apples 
in  each  row  of  the  picture  ?  Is 
each  row  separated  into  two 
parts?  How  many  apples  in 
the  two  parts  of  the  first  row  ? 
In  the  two  parts  of  the  sec- 
ond row  ?  Of  the  third  row  ? 
Fourth  row?  Fifth? 

Are  the  parts  of  the  the  first 
and   fifth  rows  the  same  ?     Of 
the  second  and  fourth?  What, 
then,   are  the  two  parts  of  G  apples?     Ans.  I  a  and  5 
a's,  2  a's  and  4  a's,  3  a's  and  3  a's. 


22  Intermediate  Arithmetic. 

(5  and  1. 

What  are  the  two  parts  of  G?     Ans.   <  4  and  2. 

1 3  and  3. 

ADDITION.  —  ~Un\tlng  the  parts, 

How  many  are  five  apples  and  1  apple?  1  apple 
and  5  apples?  4  a's  and  2  a's?  2  a's  and  4  a's? 
3  a's  and  3  a's? 

What  are  the  two  parts  of  6?  What  do  they  make 
when  united?  How  many  are  4  eggs  and  2  eggs? 

5  cows  and  1  cow?     3  baskets  and  3  baskets?     1  orange 
and  5  oranges? 

The  two  parts  of  a  set  of  chairs  are  2  chairs  and  4 
chairs;  how  many  chairs  in  the  set?  3  chickens  and 
3  chickens  are  the  parts  of  a  brood ;  how  many  chickens 
in  the  brood? 

Copy  and  read:  5  +  1  =  6;  4  +  2  =  6;  3  +  3  —  6; 
2  +  4  =  6;  1+5  =  6. 

SUBTRACTION.  —  Taking  away  one  part. 

Six  apples  are  together;  IIOAV  many  apples  would  be 
left  if  1  apple  were  taken  away  ?  If  two  apples  were 
taken  away?  If  3  apples?  4  apples?  5  apples? 

6  apples? 

What  are  the  two  parts  of  6?  When  one  of  them  is 
taken  from  6,  is  the  other  left?  A  squad  of  6  men  is 
divided  into  two  parts  ;  if  4  men  are  in  one  part,  how 
many  are  in  the  other?  6  girls  are  in  one  class;  part 
of  them  are  standing,  and  part  are  sitting ;  if  3  are  sit- 
ting, how  many  are  standing?  James  had  6  birds  in 
his  cage,  but  5  birds  got  out;  how  many  had  he  left? 

Copy  and  read:  6—1=5;  6  —  3  =  3;  6  —  5  =  1; 
6  —  2  =  4;  6  —  4  =  2;  6  —  6  =  0. 


Object  Exercises.  23 

MULT  I  PLICATION  .—Vniting  equal  parts. 

How  many  arc  1,  1,  1,  1,  1,  and  1?  How  many  times 
is  the  part  1  taken  ?  How  many,  then,  are  6  times  1  ? 
How  many  are  2,  2,  and  2?  How  many  times  is  the 
part  2  taken  ?  How  many,  then,  are  3  times  2  ?  How 
many  are  3  and  3  ?  How  many  times  is  3  taken  ? 
How  many,  then,  are  2  times  3?  Are  3  times  2  dol- 
lars .more  than  2  times  3  dollars  ? 

John  has  2  peaches,  and  Ben  has  3  times  as  many ; 
how  many  peaches  has  Ben?  Emma  has  3  tulips 
and  Rosa  has  2  times  as  many  ;  how  many  tulips  has 
Rosa  ? 

Copy  and  read :  6x1  =  6:  3x2  =  6;  2X3  =  6. 

DIVISION.—  Measuring  by  a  part. 

Does  6  contain  its  parts?  Is  6  formed  of  6  ones? 
Of  3  2's?  Of  2  3's?  How  many  times,  then,  does  6 
contain  1?  2?  3?  Make  6  a's.  Now  erase  1  a  at  a 
time.  How  many  times  did  you  erase?  How  many 
times,  then,  is  1  a  contained  in  6  a's? 

Make  6  b's.  Now  erase  2  b's  at  a  time.  How  many 
times  did  you  erase?  How  many  times,  then,  are  2 
b's  contained  in  6  b's? 

Make  6  c's.  Now  erase  3  c's  at  a  time.  How  many 
times  did  you  erase?  How  many  times,  then,  are  3 
c's  contained  in  6  c's? 

Copy  and  read  :  6  -i-  1  =  6 ;  6  -f-  2  =  3  ;  6^3  =  2. 

OBJECT  EXERCISES. 

Take  6  blocks,  put  them  on  a  table,  and  call  them 
caps. 


24  Intermediate  Arithmetic. 

What  stands  for  cap?     Ans.  c. 

Now  separate  the  caps  into  two  parts,  every  way  you 
can,  calling  the  results  thus :  1  C  and  5  c's  are  6  c's,  etc. 

Now  take  off  1  cap  at  a  time,  calling  the  result  thus ; 
1  c  from  6  c's  leaves  5  c's ;  2  c's  from  6  c's  leave  4 
c's,  etc. 

Now  put  on  1  cap  at  a  time,  calling  the  results  thus  : 
1  time  1  c  is  1  c ;  2  times  1  c  are  2  c's,  etc. 

Next  take  off  1  cap  at  a  time,  calling  the  results  thus : 
1  c  contains  1  C  1  time ;  2  c's  contain  1  c  2  times,  etc. 

Now  put  on  2  caps  at  a  time,  calling  the  results  thus : 

1  time  2  c's  is  2  c's;  2  times  2  c's  are  4  c's,  etc. 
Next  take  off  2  caps  at  a  time,  calling  the  results  thus: 

2  c's  contain  2  c's  1  time;  4  c's  contain  2  c's  2  times, 
etc. 

Now  put  on  3  caps  at  a  time,  calling  the  results  thus : 
1  time  3  c's  is  3  c's ;  2  times  3  c's  are  6  c's. 

Next  take  off  3  caps  at  a  time,  calling  the  results 
thus :  3  c's  contain  3  c's  1  time  ;  6  c's  contain  3  c's  2 
times. 

Learn  and  recite — 

THE  TABLE  OF  Six. 

0  and  6  are  6  0  from  6  leaves  6 

1  and  5  are  6  1  from  6  leaves  5 

2  and  4  are  6  2  from  6  leaves  4 

3  and  3  are  6  3  from  6  leaves  3 

4  and  2  are  6  4  from  6  leaves  2 

5  and  1  are  6  5  from  6  leaves  1 

6  and  0  are  6  6  from  6  leaves  0 

1  time    6  is     6  1  in        6,  6  times 

2  times  3  are  6  2  in        6,  3  times 

3  times  2  are  6  3  in        6,  2  times 


The  Number  tievcn. 


LESSON  VII. 

ABOUT  THE  NUMBER  SEVEN. 

What  lesson  is  this?    What  is  it  about?    How  many 
lessons   have   preceded   this?      How    many   days   in    a 
week  ?     Name   them. 
Make    7    a's    on    the 
board.     Take  7  books 
and    make   two   piles 
of   them.      What    are 
the  two  piles  called  ? 

Ans.  Parts  of  the  Avhole.     Can  you  put  6  books  in  one 
pile  and  1  book  in  the  other?     5  in  one  and  2  in  the 

other?     3  in  one  and  4  in  the  other? 

1  and  6, 

What  are  the  two  parts  of  7?      Ans.    •{  2  and  5, 

3  and  4. 


ADDITION.  —  Uniting  the  parts. 

How  many  are  6  books  and  1  book?  1  b  and  6  b's? 
2  b's  and  5  b's  ?  3  b's  and  4  b's  ?  5  b's  and  2  b's  ? 
4  b's  and  3  b's? 

What  are  the  two  parts  of  7  ?  When  you  unite  them, 
do  they  make  7?  How 'many  are  5  girls  and  2  girls? 
4  pins  and  3  pins?  1  hog  and  6  hogs?  3  mugs  and 
4  mugs? 

Ann  has  3  prunes  and  John  has  4  prunes ;  how 
many  have  both?  2  hogs  are  in  one  pen  and  5  hogs 
are  in  another;  how  many  in  both  pens? 

Copy  and  read :  6+1  —  7;  4  +  3  =  7;  2  +  5  =  7;  5 
+  2  =  =  7;  3  +  4^7;  1+6  =  7. 


26  Intermediate  Arithmetic. 

SUBTRACTION  . — Taking  away  one  part. 

Seven  books  are  together,  how  many  would  be  left 
if  1  book  were  taken  away?  If  4  books  were  taken 
away?  If  2  books?  If  6  books?  3  books?  5  books? 
7  books? 

What  are  the  two  parts  of  7  ?  When  one  of  them  is 
taken  from  7,  is  the  other  left?  How  many  are  7 
books  less  1  book?  7  walnuts  less  3  walnuts?  7  pins 
less  5  pins?  7  nuts  less  7  nuts?  7  ii's  less  2  n's?  7 
ii's  less  4  ii's?  7  11' s  less  6  ii's?  Moses  caught  7  fishes 
and  John  caught  4  fishes;  how  many  more  fishes  did 
Moses  catch  than  John  ?  Lucy  is  7  years  old  and  Betty 
is  5  ;  how  much  older  is  Lucy  than  Betty  ? 

Copy  and  read:  7--l==6;  7--3==4;  7--5==2;  7 
—7  =  0;  7  —  2=5;  7—4  =  3;  7—6  =  1. 

MULTIPLICATION.—  Uniting  equal  parts. 

How  many  are  1  +  1-fl  +  l  +  l  +  l  +  l?  How 
many  times  is  1  taken?  How  many,  then,  are  7  times 
1  ?  How  many  are  7  times  1  book  ?  Show  me  7 
planks.  Now  show  me  7  times  1  plank.  Ann  has  1 
rose  and  Mary  has  5  roses;  how  many  more  roses  does 
Mary  need  to  have  7  times  as  as  many  as  Ann?  Are 
4  r's  and  3  r's  more  than  7  times  1  r?  How  many 
do  6  cows  lack  of  being  7  times  1  cow?  How  many 
do  3  cows  lack  ? 

Copy  and  read :  7X1  =  7;  7X0^0. 

DIVISION  . — Measuring  by  a  part. 

Does  a  pile  of  7  books  contain  its  parts?  Is  1  book 
one  of  its  parts?  How  many  times  is  one  book  con- 


The  Table  of  Seven.  27 

tained  in  the  pile?     How  many  times  do  7  books  con- 
tain 1  book  ?      How  many  times  is  1  ox  contained  in 
7  oxen?     1  day  in  7  days?     1  foot  in  7  feet? 
Copy  and  read :  7-5-1  =  7. 


OBJECT  EXERCISES. 

Take  7  blocks ;  put  them  on  a  table,  and  call  them 
pies.  What  stands  for  pie?  Ans.  p. 

Now  separate  the  pies  into  two  parts,  every  way  you 
can,  calling  the  results  thus :  1  p  and  6  p's  are  7  p's, 
etc. 

Next  take  off  1  pie  at  a  time,  calling  the  results  thus  : 
1  p  from  7  p's  leaves  6  p's ;  2  p's  from  7  p's  leave 
5  p's,  etc. 

Now  put  on  1  pie  at  a  time,  calling  the  results  thus : 
1  time  1  p  is  1  p  ;  2  times  1  p  are  2  p's,  etc. 

Now  take  off  1  pie  at  a  time,  calling  the  results  thus : 
1  pie  contains  1  pie  1  time ;  2  p's  contain  1  p  2  times, 
etc. 

Learn  and  recite : 

THE  TABLE  OF  SEVEN. 

0  and  7  are  7  0  from  7  leaves  7 

1  and  6  are  7  1  from  7  leaves  6 

2  and  5  are  7  2  from  7  leaves  5 

3  and  4  are  7  3  from  7  leaves  4 

4  and  3  are  7  4  from  7  leaves  3 

5  and  2  are  7  5  from  7  leaves  2 

6  and  1  are  7  6  from  7  leaves  1 

7  and  0  are  7  7  from  7  leaves  0 

7  times  1  are  7  1  in       7,  7  times. 


28 


Intermediate  Arithmetic. 


LESSON  VIII. 

ABOUT  THE  NUMBER  EIGHT. 

^ 

What  lesson  is  this?    What  is  it  about?     How  many 

lessons  have  preceded 
this?  In  the  picture 
is  a  squad  of  soldiers ; 
h  o  w  many  soldiers 
are  in  the  whole 
squad  ?  If  the  squad 
were  divided  into  two 
parts,  how  many  sol- 
diers would  be  in  each 
part?  Ans.  1  soldier  in  one  part  and  7  soldiers  in  the 
other,  or  2  soldiers  and  6  soldiers,  or  3  soldiers  and  5 
soldiers,  or  4  soldiers  and  4  soldiers.  How  many  sets 
of  two  parts  are  there?  Take  8  books;  call  them  sol- 
diers, and  separate  them  into  4  sets  of  two  parts. 

What  are  the  two  parts  of  8?     Am.  j  1  and  7>  3  and  5' 

(.  2  and  6,  4  and  4. 

ADDITION.  —Uniting  the  parts. 

How  many  are  7  soldiers  and  1  soldier?  1  s  and 
7  S's  ?  3  s's  and  5  s's  ?  6  s's  and  2  s's  ?  4  birds  and 
4  birds?  5  b's  and  3  b's?  2  b's  and  6  b's? 

What  are  the  two  parts  of  8?  What  do  they  make 
when  united  or  put  together?  How  many  are  5  marbles 
and  3  marbles?  4  ill's  and  4  ill's?  6  cakes  and  2 
cakes?  7  buds  and  1  bud?  How  manv  is  1  more  than 

ti 

7  ?  3  more  than  5  ?  4  more  than  4  ?  Ike  has  5  peaches, 
and  Jim  has  three  peaches  more  than  Ike;  how  many 
peaches  has  Jim?  Ben  jumped  6  feet  and  Tom  jumped 
2  feet  further  than  Ben;  how  far  did  Tom  jump? 


The  Number  Eight.  29 

Copy  and  read :  1  +  7  =  8;  3  +  5  =  8;  5  +  3  =  8; 
7_|_1==8;  2  +  6  =  8;  4  +  4  =  8;  6  +  2  =  8;  8  +  0  =  8. 

SUBTRACTION.  —  Taking  away  one  part. 

How  many  soldiers  would  be  left  in  the  squad  if  1 
soldier  were  taken  away  ?  If  two  soldiers  were  taken 
away?  If  3  soldiers?  If  4  soldiers?  5  soldiers?  6 
soldiers  ?  7  soldiers  ?  8  soldiers  ? 

What  are  the  two  parts  of  8?  When  one  of  them  is 
taken  from  8,  what  is  left?  I  wish  to  put  8  birds  in 
two  cages  so  as  to  have  3  birds  in  one  cage,  how  many 
birds  will  be  in  the  other  cage?  How  can  I  put  8  hogs 
in  two  pens  so  as  to  have  2  hogs  in  one  of  the  pens? 
Moses  and  Joseph  ate  8  biscuits  together  ;  if  Joseph  ate 
4  biscuits,  how  many  did  Moses  eat?  Which  is  the 
more,  8  or  5  ?  How  many  more  ? 

Copy  and  read :  8  —  2  =  6;  8  —  3=5;  8  —  6  =  2. 

MULTIPLICATION.  —  Uniting  equal  parts. 

Make   8   marks.     How   many   times  did  you  make  1 

«/  *' 

mark?  8  times  1  mark  are  how  many?  Are  2  marks 
a  part  of  8  marks  ?  If  you  make  2  marks  4  times,  thus : 
//,  //,  I  It  111  how  many  marks  will  there  be  in  all? 
How  man}7,  then,  are  4  times  2  marks?  4  times  2  sol- 
diers? Are  4  marks  a  part  of  8  marks?  If  you  make 
4  marks  2  times,  thus:  ////,  ////,  how  many  marks 
will  there  be  in  all  ?  How  many,  then,  are  2  times  4 
marks?  2  times  4  soldiers? 

Ike  has  4  marbles  and  John  has  2  times  as  many  as 

«. 

Ike  ;  how  many  marbles  has  John  ?  Harry  has  4  times 
2  dimes  and  Jane  has  2  times  4  dimes  which  has  the 
more  ? 

Copy  and  read  :  8x1     -8;  4x2=^8;  2  X  4  =  =  8. 


30  Intermediate  Arithmetic. 

DIVISION.  —  Measuring  by  a  part. 

Does  8  contain  its  parts  ?  Does  the  squad  of  8  soldiers 
contain  1  soldier  8  times?  Suppose  2  soldiers  were 
taken  away ;  then  2  soldiers  more  were  taken  away ; 
then  2  more;  then  2  more;  would  there  be  any  soldiers 
left?  How  many  times  can  2  soldiers  be  taken  away? 
How  many  times,  then,  do  8  soldiers  contain  2  soldiers? 
If  4  soldiers  were  taken  away,  and  then  4  soldiers  more, 
would  any  soldiers  remain  ?  How  many  times  can  4 
soldiers  be  taken  away?  How  many  times,  then,  do  8 
soldiers  contain  4  soldiers? 

OBJECT  EXERCISES. 

Take  8  blocks ;  put  them  on  a  table  and  call  them 
soldiers.  What  stands'  for  soldier  ?  Ans.  S. 

Now  separate  the  soldiers  into  two  parts  every  way 
you  can,  calling  the  results  thus :  1  s  and  7  s's  are  8  s's, 
etc. 

Now  take  off  1  soldier  at  a  time,  calling  the  results 
thus  :  1  s  from  8  s's  leaves  7  s's ;  2  s's  from  8  s's  leave 
6  s's,  etc. 

Now  put  on  1  soldier  at  a  time,  calling  the  results 
thus  :  1  time  1  s  is  1  s ;  2  times  1  s  are  2  s's,  etc. 

Now  take  off  1  s  at  a  time,  calling  the  results  thus  : 

1  s  contains  1  s,  1  time;  2  s's  contain  1  s,  2  times,  etc. 
Next  put  on  2   soldiers  at  a  time,  calling  the  results 

thus  :    once  2  s's  is  2  s's ;    2  times  2  s's  are  4  s's,  etc. 

Next  take  off  2  soldiers  at  a  time,  calling  the  results 

thus :  2  s's  contain  2  s's,  1  time ;  4  s's  contain  2  s's,  2 

2  times,  etc. 

Now  put  on  4  soldiers  at  a  time,  etc.,  and  then  take 
off  4  s's  at  a  time,  etc. 


The  Number  Nine. 


31 


Learn  and  recite : 

THE  TABLE  OF  EIGHT. 


0  and      8  are  8 

0  from 

1  and      7  are  8 

1  from 

2  and      6  are  8 

2  from 

3  and      5  are  8 

3  from 

4  and      4  are  8 

4  from 

5  and      3  are  8 

5  from 

6  and      2  are  8 

6  from 

7  and      1  are  8 

7  from 

8  and      0  are  8 

8  from 

1  time    8  is     8 

1  in 

2  times  4  are  8 

2  in 

4  times  2  are  8 

4  in 

8  leaves  8 

8  leaves  7 

8  leaves  6 

8  leaves  5 

8  leaves  4 

8  leaves  3 

8  leaves  2 

8  leaves  1 

8  leaves  0 

8,  8  times 
8,  4  times 
8,  2  times 


LESSON  IX. 

ABOUT  THE  NUMBER  NINE. 

Is  9  one  more  than  8  ?     Hold  up  9  fingers.   •  Make  9 
steps  and  count  them  as  you  step. 


clclilclciJJclclci 


How  many  a's  are  in  the  above  row  ?  Count  and 
see  if  there  are  8  a's  and  la?  7  a's  and  2  a's?  6  a's 
and  3  a's?  5  a's  and  4  a's? 

What  are  the  two  parts  of  9?  An*.  /  ij  an(J  *'  7  and  2< 

(>  and  3,  5  and  4. 


32  Intermediate  Arithmetic. 

ADDITION  .—  Uniting  the  parts. 

Count  9.  How  many  are  8  men  and  1  man  ?  7  m's 
and  2  ill's?  6  cats  and  3  cats?  5  c's  and  4  c's? 
What  are  the  two  parts  of  9?  When  united  do  they 
make  9  ?  Is  1  more  than  8  the  same  as  2  more  than 
7?  Is  3  more  than  6  the  same  as  4  more  than  5? 
How  many  are  4  and  5  ?  5  and  4  ?  6  and  3  ?  3  and 
6?  7  and  2?  2  and  7? 

Five  eggs  in  one  nest  and  4  eggs  in  another;  how 
many  eggs  in  all  ?  7  tulips  and  2  tulips  are  how 
many  ?  6  daisies  in  one  cluster  and  3  in  another ; 
how  many  daisies  in  both  clusters  ? 

Copy  and  read:  1+8  =  9;  2+7=9;  3  +  6  =  9;  5  + 
4  =  9. 

S  U  BTR  ACTION  .  —  Taking  aivay  one  part. 

Count  9  backward.  How  many  are  2  ducks  less  than 
9  ducks?  4  doves  less  than  9  doves?  6  rats  less  than 
9  rats?  What  are  the  two  parts  of  9?  When  one  of 
them  is  taken  from  9,  what  is  left?  9  birds  were  on  a 
limb,  but  three  of  them  have  flown,  how  many  birds 
are  left?  The  old  hen  has  9  chicks;  6  of  the  chicks 
are  on  one  side  of  the  fence,  how  many  are  on  the  other 
side?  A  boy  had  9  marbles  but  lost  5  of  them,  how 
many  marbles  had  he  left?  If  7  chairs  are  1  part  of  a 
set  of  9  chairs,  how  many  are  in  the  other  part?  7 
from  9  leaves  how  many  ? 

Copy  and  read:  9  —  1  =  8;  9—2  =  7;  9  —  3  =  6;  9  — 
4  =  5;  9-5=4. 

MULTIPLICATION.-  Uniting  equal  parts. 

Make  9  aV.  How  many  times  did  you  make  la? 
How  many,  then,  are  9  times  l*a?  9  times  1  horse? 


Object  Exercises.  33 

9  times  1  boy?  Are  3  a's  a  part  of  9  a's?  If  you 
make  3  a's  3  times,  thus  :  aaa,  aaa,  aaa,  how  many 
a's  will  there  be  in  all?  How  many,  then,  are  three 
times  3?  3  times  3  cows?  3  times  3  cats?  Show  me 
3  times  3  fingers.  Show  me  3  times  3  planks.  Little 
Hal  is  3  years  old,  and  Mattie  is  three  times  as  old  as 
Hal;  how  old  is  Mattie?  Are  9  times  1  dollar  more 
than  3  times  3  dollars? 

Copy  and  read  :  9X1  =  9;  3X3  =  9. 

DIVISION  .—Measuring  by  a  part. 

Does  9  a  contain  its  parts?  Is  1  a  one  of  the  parts? 
How  many  a's  does  a  row  of  9  a's  contain  ?  Make  9 
a's.  Erase  3  a's  at  a  time,  until  all  are  gone.  How 
many  times  did  you  erase?  How  many  times,  then, 
are  3  a's  contained  in  9  a's? 

Is  a  yard -stick  3  feet  long?  How  many  times  is  a 
yard -stick  contained  in  9  feet?  What  is  a  yard -stick 
for?  Ans.  For  measuring.  Can  I  measure  a  pole,  9 
feet  long,  by  it?  How  many  measures  would  it  take? 
How  many  does  7  lack  of  containing  one  9  times? 
How  many  does  5  lack  ?  6  ?  8  ?  3  ?  How  many  do  8 
cups  lack  of  containing  3  cups  3  times?  How  many 
do  4  cups  lack? 

Copy  and  read:  9-^-1=9;  9^3  =  3;  9  -=-  9  =  1. 

OBJECT  EXERCISES. 

Take  9  blocks,  put  them  on  the  table  and  call  them 
horses.  What  stands  for  horses?  Ans.  li. 

Now  separate  the  horses  into  two  parts  every  way 
you  can,  calling  the  results  thus  :  1  li  and  8  li's  are  9 
h's,  etc. 

N.  I.— 3. 


34  Intermediate  Arithmetic. 

Next  take  off  1  horse  at  a  time,  calling  the  results 
thus:  1  horse  from  9  horses  leaves  8  horses;  2  li's  from 
9  h's  leave  7  h's,  etc. 

Next  put  on  one  horse  at  a  time,  calling  the  results 
thus:  1  time  1  h  is  1  h;  2  times  1  h  are  2  h's,  etc. 

Next  take  off  1  horse  at  a  time,  calling  the  results 
thus:  1  h  contains  1  h,  1  time;  2  h's  contain  1  h,  2 
times,  etc. 

Next  put  on  3  horses  at  a  time,  calling  the  results 
thus :  1  time  3  h's  is  3  h's ;  2  times  3  h's  are  6  h's,  etc. 

Now  take  off  3  h's  at  a  time,  calling  the  results  thus : 
3  h's  contain  3  h's,  1  time ;  6  h's  contain  3  h's,  2  times, 
etc. 

NOTE.— The  teacher  may  repeat  the  object  exercises,  giving 
the  blocks  or  pebbles  such  names  as  will  interest  and  amuse 
the  pupils. 

Learn  and  recite- 

THE  TABLE  O,F  NINE. 

0  and  9  are  9  0  from  9    leaves  9 

1  and  8  are  9  1  from  9    leaves  8 

2  and  7  are  9  2  from  9    leaves  7 

3  and  6  are  9  3  from  9    leaves  6 

4  and  5  are  9  4  from  9    leaves  5 

5  and  4  are  9  5  from  9    leaves  4 

6  and  3  are  9  6  from  9    leaves  3 

7  and  2  are  9  7  from  9    leaves  2 

8  and  1  are  9  8  from  9    leaves  1 

9  and  0  are  9  9  from  9    leaves  0 

1  time     9  is     9  1  in       9,  9  times 

3  times   3  are  9  3  in       9,  3  times 


The  Number  Ten. 


35 


LESSON  X. 

ABOUT  THE  NUMBER  TEN. 

What  lesson  is  this?  What  is  it  about?  10  is  the 
next  number  after  what?  Count  all  your  fingers.  How 
many  have  you  ?  How 
many  frogs  in  the  pic- 
ture? How  many  are 
on  the  log?  How 
many  are  on  the 
ground  ?  If  another 
frog  gets  on  the  log, 
how  many  would  then 
be  on  the  log  and 
ground  ?  How  many 
if  2  frogs  more  get  on  the  log?  How  many  if  3  frogs 
more  get  on  the  log?  4  frogs  more?  5  frogs?  Is  one 
part  of  10  frogs  on  the  log  and  the  other  part  on  the 
ground  ?  (  1  and  9,  3  and  7, 

What  are  the  two  parts  of  10?   Ans.  <  2  and  8,  4  and  6, 

15  and  5. 

ADD  IT  ION.  —  Uniting  the  parts. 

How  many  are  7  frogs  and  3  frogs?     8  f's  and  2  f's? 
9  f's  and  1  f  ?     4  men  and  6  men  ?    5  ill's  and  5  in's  ? 

3  birds  and  7  birds?     2  b's  and  8  b's?     What  are  the 
two  parts  of  10?     How  many  do  they  make  when  put 
together?     How   many  are  5   marbles  and  5   marbles? 
7  eggs  and  3  eggs?     6  cakes  and  4  cakes? 

Charles  is  7  years  old  and  Fannie  is  3  years  older; 
how  old  is  Fannie?     Mary  has  6  roses  and  Jane  has 

4  roses;  how  many  roses  have  both  girls? 

Copy  and  read  :  4  -f  6  =  10;  8  -f  2  =  10;  3  -f  7  =  10. 


36  Intermediate  Arithmetic. 

SUBTRACTION.  —  Taking  away  one  part. 

If  10  frogs  were  on  a  log,  and  1  frog  should  leap  off, 
how  many  frogs  would  be  left?  How  many  would  be 
left  if  2  frogs  should  leap  off?  If  3  frogs  should  leap 
off?  If  4  frogs?  5  frogs?  6  frogs?  7  frogs?  8  frogs? 
9  frogs?  30  frogs? 

What  are  the  two  parts  of  10?  When  one  of  them 
is  taken  from  10,  what  is  left?  How  many  are  8  and 
2  ?  How  many,  then,  is  10  less  2  ?  There  are  10  frogs 
in  all ;  if  4  frogs  are  on  the  log,  how  many  are  on  the 
ground  ? 

Ann  had  10  little  birds,  but  the  old  cat  ate  3  of 
them ;  how  many  birds  has  Ann  now  ?  10  pinks  in 
one  bed,  and  6  pinks  hi  another  bed ;  how  many  more 
pinks  in  one  than  in  the  other?  10  hogs  were  in  the 
garden,  but  John  turned  4  hogs  out;  how  many  re- 
main in  the  garden  ? 

Copy  and  read:  10  —  3  =  7;  10  —  5  =  5;  10  —  8  =  2. 

MULTIPLICATION.—  Uniting  equal  parts. 

Count  10.  Make  10  c's.  How  many  times  did  you 
make  1  c?  How  many,  then,  are  10  times  1  c?  10 
times  1  horse?  Are  2  c's  a  part  of  10  c's?  If  you 
make  2  c's  5  times,  thus  :  CC,  CC,  CC,  cc,  CC ;  how 
many  c's  will  there  be  in  all?  How  many,  then,  are  5 
times  2  c's  ?  5  times  2  cups  ?  5  times  2  caps  ?  If  you 
make  5  c's  2  times,  how  many  c's  will  there  be  in  all? 
How  many,  then,  are  2  times  5  c's?  2  times  5  cats? 
Are  5  times  2  dollars  the  same  as  2  times  5  dollars? 
Little  Ella  is  5  years  old,  and  her  brother  is  2  times 
as  old  as  she;  how  old  is  her  brother? 

Copy  and  read  :  10  X  1  =  10  ;  5  X  2  =  -- 10 ;  2  X  5  =  10. 


Object  Exer  rises  *  37 

DIVISION.  —  Measuriny  by  <i  part. 

Does  10  contain  its  parts?  How  many  1's  in  10? 
How  many  times,  then,  does  10  contain  1  ?  Make  10 
c's.  Now  erase  2  c's  at  a  time  until  all  are  gone;  how 
many  times  did  you  erase?  How  many  times,  then, 
do  10  c's  contain  2  c's?  10  caps  contain  2  caps?  10 
cabs  contain  2  cabs?  Show  me  10  fingers.  Now  re- 
move 5  fingers  at  a  time;  how  many  times  did  you 
remove?  How  many  times,  then,  do  10  fingers  contain 
5  fingers?  10  fobs  contain  5  fobs?  10  fans  contain  5 
fans  ? 

Copy  and  read  :  10  -r- 1  =  10 ;  10^-2=5;  10  ~  5  =  2. 

OBJECT  EXERCISES. 

Put  10  blocks  on  a  table  and  call  them  lions.  What 
stands  for  lions?  Ans.  1. 

Now  separate  the  lions  into  two  parts  every  way  you 
can,  calling  the  results  thus :  1  1  and  9  1's  are  10  1's, 
etc. 

Now  take  off  1  lion  at  a  time,  calling  the  results  thus : 
1  1  from  10  1's  leaves  9  1's;  2  1's  from  10  1's  leave  8 
1's,  etc. 

Now  put  on  1  lion  at  a  time,  calling  the  results  thus : 
1  time  1  1  is  1  1;  2  times  1  1  are  2  1's,  etc. 

Next  take  off  1  lion  at  a  time,  calling  the  results  thus : 
1  1  contains  1  1,  1  time;  2  1's  contain  1  1,  2  times,  etc. 

Next  put  on  2  lions  at  a  time,  calling  the  results 
thus :  1  time  2  1's  is  2  1's  ;  2  times  2  1's  are  4  1's,  etc. 

Now  take  off  2  lions  at  a  time,  calling  the  results 
thus :  2  1's  contain  2  1's  1  time ;  4  1's  contain  2  1's  2 
times,  etc. 


38  Intermediate  Arithmetic. 

THE  TABLE  OF  TEN. 

0  and  10  are  10  0  from  10  leaves  10 

1  and  9  are  10  1  from  10  leaves  9 

2  and  8  are  10  2  from  10  leaves  8 

3  and  7  are  10  3  from  10  leaves  7 

4  and  6  are  10  4  from  10  leaves  6 

5  and  5  are  10  5  from  10  leaves  5 

6  and  4  are  10  6  from  10  leaves  4 

7  and  3  are  10  7  from  10  leaves  3 

8  and  2  are  10  8  from  10  leaves  2 

9  and  1  are  10  9  from  10  leaves  1 

1  time    10  is     10  1  in       10,  10  times 

2  times    5  are  10  2  in       10,  5     times 
5  times    2  are  10                5  in       10,  2    times 

REVIEW  QUESTIONS. 

MENTAL  EXERCISES. 

What  is  Addition?    Ans.  Uniting  parts  or  numbers. 

What  is  the  sign  of  Addition?  Ans.  +.  What  is  it  called? 
Ans.  And. 

What  is  the  number  whose  two  parts  are  4  and  3?  1  and 
6?  2  and  5?  7  and  3?  2  and  2?  1  and  8?  3  and  2?  2  and 
7?  5  and  4?  2  and  6?  6  and  3?  5  and  4?  3  and  3?  4  and 
2?  3  and  5?  8  and  2?  4  and  4?  5  and  5?  6  and  4? 

How  many  are  4  birds  and  3  birds?  1  b  and  2  b's?  2  b's 
and  5  b's?  7  cats  and  3  cats?  2  c's  +  7  c's?  4  hats  and  4 
hats?  2  h's  and  6  h's?  4  men  and  2  men?  3  m's  and  5  m's? 

What  is  Subtraction  ?  Ans.  Taking  away  one  part  of  a  num- 
ber, or  taking  one  number  from  another. 

What  is  the  sign  of  Subtraction  ?  Ans.  --.  What  is  it  called? 
Ans.  Less. 

Two  is  one  part  of  the  number  3,  what  is  the  other  part? 
What  is  the  other  part  of  7  if  3  is  one  part?  If  6?  What 


Review  Questions.  39 

is  the  other  part  of  9  if  5  is  one  part?  If  7?  If  3?  What  is 
the  other  part  of  8  if  6  is  one  part  ?  If  3  ?  If  4  ? 

How  many  are  7  birds  less  3  birds  ?  7  b's  less  5  b's  ?  9  caps 
less  1  cap?  8  c's  — 6  c's?  9  tubs  less  3  tubs?  6  t's  less  3 
t's?  10  fans  less  3  fans?  lOf's--  5  f's? 

What  is  the  difference  between  7  and  2?  10  and  3?  9  and 
4  ?  8  marbles  and  5  marbles  ?  5  m's  and  3  ni's  ?  10  rats  and 
8  rats?  9  rats  and  5  rats?  7  r's  and  4  r's? 

What  is  Multiplication  ?  Ans.  Uniting  equal  parts  or  num- 
bers. What  is  the  sign  of  Multiplication?  Ans.  X-  What  is  it 
called?  Ans.  Times. 

How  many  are  1  taken  5  times?  1  taken  6  times?  3  taken 
2  times  ?  2  taken  3  times  ?  5  taken  2  times  ?  2  taken  5  times  ? 
2  taken  2  times  ?  4  taken  2  times  ?  2  taken  4  times  ?  0  taken 
7  times?  7  taken  0  times? 

How  many  are  9  times  1  bird?  7  times  1  b?  1  time  5 
birds?  6  times  1  hat?  3X2  h's?  10  times  1  box?  5X2  b's? 
3X3  b's?  4X2  b's?  7  times  1  watch ?  2X4w's?  2X2w's? 

What  is  Division?  Ans.  Measuring  a  number  by  one  of  its 
parts,  or  by  another  number.  What  is  the  sign  of  Division? 
Ans.  -5-.  What  is  it  called?  -4ns.  Contains. 

How  many  times  does  5  contain  1  ?  7  contain  1  ?  10  con- 
tain 2  ?  8  contain  4  ?  6  contain  2  ?  4  contain  2  ?  6  contain  6  ? 
G  contain  3?  8  contain  2?  10^5?  9n-3? 


SLATE  EXERCISES. 

Copy  and  add — 

122312345633 
112245623274 

112323567863 
231220131143 
356547312003 


40  Intermediate  Arithmetic. 


2 

1 

1 

3 

2 

4 

1 

3 

2 

1 

1 

1 

2 

3 

0 

1 

3 

1 

0 

3 

2 

4 

1 

0 

0 

1 

4 

1 

1 

3 

6 

3 

2 

4 

2 

3 

G 

2 

4 

5 

1 

0 

3 

1 

2 

1 

6 

5 

Copy  and  subtract- 

2 

3 

4 

5 

5 

7 

9 

6 

8 

5 

4 

7 

1 

2 

2 

'3 

1 

3 

5 

4 

3 

2 

1 

5 

7  10   968579  10   869 
263324447626 

9   6   8  10   7   9  10   8   6  10   9   8 
765568375224 

Copy  and  multiply — 

532478142135 
112111723832 

292317810524 
514260149232 

Copy  and  divide — 

1)5    2)6    7)7    2)4     3)6    2)10     1)9     6)6     1)10    2)8 

1)8    4)8    5)5    3)3    2)6    2)8    2)10    3)9    4)8    5)10 


PART  II. 


NOTATION  AND  NUMERATION. 


1.  A  Unit  is  a  single  thing  or  one ;  as  one,  one  dime, 
one  dozen,  one  hundred. 

2.  A  Number   is  a   unit  or  a  collection  of  units;    as 
one,  five,  seven  men,  twenty  days. 

3.  The  Unit  of  a  Number  is  one  of  the  units  of  which 
the  number  is  formed.     Thus,  the  unit  of  nine  is  one; 
of  twelve  quarts,  one  quart;   of  eleven  dozen,  one  dozen. 

4.  An  Abstract  Number  is  a  number  that   is  not  ap- 
plied to  any  particular  object;  as  three,  ten,  fifteen. 

5.  A  Concrete  Number  is  a  number  that  is  applied  to 
some  particular  object ;  as  two  cents,  six  bushels,  twenty- 
five  dollars. 

6.  EXERCISES.  —  What    is    the    unit   of   Four?     Five 
boys?     Nine?     Twelve  dollars?     Twenty  days?     Which 
of  these  numbers  are  abstract  and  which  are  concrete? 
Name  five  abstract,  also  five  concrete  numbers. 

7.  Notation    is    the    art    of    expressing    numbers    by 
symbols. 

8.  Numeration    is   the   art   of   reading    numbers    that 
are  expressed  by  symbols. 

9.  Symbols  may  be  either  letters  or  figures. 

(41) 


42 


Intermedia te  A r itlm letic. 


NUMBERS  FROM  0  TO  10. 

10.   In   the  Arabic   Notation   ten   figures   are   used   to 
express  numbers,  viz: 


Figures  -0,      1,    2,      3,     4,     5,    6,     7,      8,     9. 

Names  —  Naught,  One,  Two,  Three,  Four,  Five,  Six,  Seven,  Eight,  Nine. 

11.  The  figure  0,  also   called   cipher   or  zero,  denotes 
none   or   nothing.      The    other    nine    figures    are    called 
digits. 

12.  The   largest  number   that  can  be   expressed  by  a 
single  figure  is  nine  or  9.     Nine  and  one  are  ten,  which 
is  written  10. 


NUMBERS  FROM  10  TO  100. 


a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

10th  Row. 
9th   " 
8th   " 
7th   " 
6th   " 
5th   " 
4th   " 
3d   " 
2d   " 
1st   " 

a 
a 

a 

a 

a 

a 

a 

a 

a 

Column  of  Tens. 


Column  of  Units. 


Notation  and  Numeration. 


43 


13.  How  many  a's  in  each  row   of  units?     Ans.  one. 
How  many  a's  in  each  row  of  tens?  Ans.  1  ten. 
How  many  a's  in  2  rows  of  tens?             Ans.  2  tens. 
How  many  a's  in  3  rows  of  tens  ?    In  4  rows  of  tens  ? 
In  5  rows  of  tens?     In  6?     In  7?     8?    9?     10? 
Count  the  a's  in  the  first  2  rows  of  tens;  in  the  first 

3  rows  of  tens;  in  the  first  4  rows;  in  the  first  5  rows, 
eic.,  eic. 

How  many  tens  in  twenty?  In  thirty?  Forty? 
Fifty?  Sixty?  Seventy?  Eighty?  Ninety?  One 
hundred  ? 

How  many  units  in  2  tens?  In  3  tens?  In  4  tens? 
Etc.,  etc. 

Count  the  a's  in  2  rows  of  tens  and  7  rows  of  units. 
What  is  the  short  method  of  denoting  the  number  of 
a's  in  "  2  rows  of  tens  and  7  rows  of  units  ? '  Ans. 
27  a's.  Point  out  the  rows  that  you  would  count  to 
get  42  a's ;  57  a's ;  63  a's ;  91  a's ;  35  a's ;  17  a's ;  20  a's. 

14.  When  a  number  is  expressed  by  two  figures,  the 
figure  at  the  right  denotes  units  or  ones,  and  the  figure 
at  the  left  denotes  tens. 

By  counting,  show  that : 

16  stands  for  Sixteen. 

"    Seventeen. 
"    Eighteen. 
Nineteen. 
"    Twenty. 
"    Twenty-one. 

What  do  the  figures  57  denote? 

Ans.  5  tens  7  ones ;  read  fifty-seven. 
What  do  the  figures  83  denote? 

Ans.  8  tens  3  ones ;  read  eighty-three. 


10  stands  for  Ten. 

16 

11 

"    Eleven. 

17 

12 

"    Twelve. 

18 

13       " 

"    Thirteen. 

19 

14 

"    Fourteen. 

20 

15      " 

"    Fifteen. 

21 

a 


44 


Intermediate  Arithmetic. 


15,  Copy  and  read  the  following  numbers,  naming  the 
tens  and  units  in  each : 


23. 

47. 

13. 

63. 

73. 

36. 

19. 

42. 

58. 

24. 

35. 

78. 

27. 

33. 

71. 

85. 

78. 

40. 

99. 

31. 

55. 

93. 

60. 

81. 

49. 

88. 

69. 

100. 

16.  Express  by  figures : 


1.  Seventy-three. 

2.  Sixty-nine. 

3.  Eighty-eight. 
Forty-nine. 
Ninety-nine. 
Fifty-six. 
Eighty-three. 
Forty-four. 


9.  Seventy-seven 

10.  Ninety. 

11.  Seventy-eight. 

12.  Thirty-nine. 

13.  Ninety-six. 

14.  Eighty-four. 

15.  Fifty-nine. 


17.  Twenty-seven. 

18.  Forty-five. 

19.  Seventy-one. 

20.  Sixty-seven. 

21.  Eighty -one. 

22.  Sixty-six. 

23.  Twenty-nine. 

24.  One-hundred. 


16.  Forty-seven. 

17.  The   greatest  number  that  can  be  expressed  by 
two  figures  is  ninety-nine,  or  99,  which  represents  9  tens 
and  9  units.     99  and  1  more  make  one  hundred,  written 
100.     100  is  10  tens. 

NUMBERS  FROM  100  TO  1000. 

18.  Ten  rows  of  ten  a's,  arranged  as  on  page  42,  rep- 
resent the  numeral  frame.     How  many  a's  are  in  one 
frame?       Ans.  One  hundred. 

How  many  in  2  frames?     Ans.  Two  hundred. 

How  many  in  3  frames?  In  4  frames?  In  5  frames? 
etc.  How  many  a's  are  in  3  frames  5  rows  of  tens  and 
7  rows  of  ones?  Ans.  3  hundreds  5  tens  7  ones,  which  is 
written  357,  and  read  three  hundred  fifty-seven. 

19.  When  a  number  is  expressed  by  three  figures,  the 
figure  at  the  right  denotes  units ;  the  middle  figure,  tens; 
and  the  figure  at  the  left,  hundred*. 


Notation  and  Numeration. 


45 


One  hundred,     or  1  hd.  0  tens,  0  ones,  is  denoted  by  100 

Two  hundred, 

Three  hundred, 

Four  hundred, 

Five  hundred, 

Six  hundred, 

Seven  hundred, 

Eight  hundred, 

Nine  hundred,    "  9 


a 

2 

a 

0 

a 

0 

" 

3 

« 

0 

It 

0 

it 

4 

it 

0 

a 

0 

tt 

5 

" 

0 

" 

0 

tt 

6 

tt 

0 

it 

0 

" 

7 

tt 

0 

tt 

0 

ti 

8 

" 

0 

« 

0 

tt 

9 

tt 

0 

tt 

0 

tt 


It 


ti 


it 


tt 


It 


It 


a 

200 

It 

300 

It 

400 

It 

500 

tt 

600 

tt 

700 

tt 

800 

It 

900 

What  do  the  figures  345  denote? 


Ans.  3   hund. 


4  tens  5  ones ;  read,  three  hundred  forty-five. 

20.  Read  the  following  numbers,  naming  the  hundreds, 
tens,  and  units  of  each: 


107. 

538. 

297. 

580. 

777. 

896. 

756. 

627. 

268. 

579. 

180. 

403. 

865. 

921. 

354. 

412. 

141. 

501. 

494. 

330. 

945. 

813. 

754. 

999. 

21.  Express  by  figures: 

1.  Two  hundred  twenty-two. 

2.  Three  hundred  thirty-three. 

3.  Eight  hundred  five. 

4.  Five  hundred  twenty-seven. 

5.  Seven  hundred  forty-one. 

6.  Nine  hundred  seventy-four. 

7.  Four  hundred  ninety-nine. 

8.  Six  hundred  sixty-six. 

9.  Four  hundred  fifty. 

10.  Nine  hundred   eighty-one. 

11.  Seven  hundred  sixty-seven. 

12.  Six  hundred  thirty-two. 

13.  One  hundred  two. 


46 


Intermediate  A rithmetic. 


14.  Three  hundred  fifteen. 

15.  Two  hundred  twelve. 

16.  Nine  hundred  ninety-nine. 

22.  The   greatest   number   that   can   be  expressed  by 
three  figures  is  nine  hundred  ninety-nine,  or  999,  which 
represents   9   hunds.    9  tens  9  units.      999    and   1    more 
make  one  thousand ;  written,  1000.    1000  is  10  hundreds. 

NUMBERS  FROM  1000  TO  10000. 

23.  If  we  call  ten  "numeral  frames"  &  pack,  how  many 
a's  are  in  one  pack?  Ans.  One  thousand. 

How  many  a's  in  2  packs?          Ans.   Two  thousand. 

How  many  in  3  packs?  In  4  packs?  In  5  packs? 
etc. 

How  many  a's  in  7  packs,  3  frames,  2  rows  of  tens, 
and  5  rows  of  ones? 

Ans.  7  thousands  3  hundreds  2  tens  5  ones,  which 
is  written  7325,  and  read,  seven  thousand  three  hundred 
twenty-five. 

What  do  the  figures  4609  denote? 

Ans.  4  thousands  6  hundreds  0  tens  9  ones;  read, 
four  thousand  six  hundred  nine. 

What  do  the  figures  7040  denote  ? 

-4ns.  7  thousands  0  hundreds  4  tens  0  ones ;  read,  seven 
thousand  forty. 

What  do  the  figures  9003  denote  ? 

Ans.  9  thousands  0  hundreds  0  tens  3  ones ;  read,  nine 
thousand  three. 

24.  Copy  and  read : 


1974. 

3541. 

4200. 

7532. 

8007. 

8013. 

5727. 

2975. 

5380. 

6483. 

6304. 

7090. 

9756. 

3868. 

9999. 

Notation  and  Numeration.  47 

25,  Express  by  figures : 

1.  One  thousand  four  hundred  seventy-five. 

2.  Three  thousand  two  hundred  nineteen. 

3.  Five  thousand  seven  hundred  twenty-seven. 

4.  Seven  thousand  three  hundred  fifty-four. 

5.  Two  thousand  six  hundred  twelve. 

6.  Eight  thousand  one  hundred  forty. 

7.  Nine  thousand  nine  hundred  ninety-nine. 


8.  Nine  thousand  six. 

9.  Four  thousand  ten. 
10.  Six  thousand  eleven. 


11.  One  thousand  fifteen. 

12.  Five  thousand  fifty. 

13.  Seven  thousand  one. 


WRITING  AND  READING  NUMBERS  IN  GENERAL. 

PLACES  AND  PERIODS. 

26.  In  any  number  the  places  or  orders  of  the  figures 
are  numbered  from  the  right.     Thus,  in  3425867, 


7  is  in  the  1st  place, 
6  "   "     "    2d      li 

8  "  "  "  3d   " 
5  "  "  "  4th  " 


2  is  in  the  5th  place, 
4  "  "  "  6th   u 

3  "  "  "  7th  " 

etc.,  etc. 


27.  The  periods,   which  consist  of  three  figures  each, 
are  also  numbered  from  the  right.     Thus,  in  345,896,701, 

701  is  the  1st  period. 

896  "     "     2d         " 
345  "     "     3d        " 

EXERCISES. 

28.  In  42375809,  what  figure  is  in  the  2d  place  ?     In 
the  5th  ?     7th  ?     1st  ?     3d  ?     Of  what  order  is  8  ? 

Ans.  3d  order. 


48  Intermediate  Arithmetic. 

Mention  the  order  of  each  figure.  Separate  the  num- 
ber into  periods  by  commas.  Which  is  the  1st  period? 
The  2d?  3d?  How  many  figures  in  the  3d  period? 
What  is  the  unit  of  7?  9  cents?  5  tern?  What  is 
the  value  of  5  if  the  unit  is  one  peck  ?  Ans.  5  pecks. 

What  is  the  value  of  5  if  the  unit  is  1  ten? 

Ans.  5  tens  or  50. 

What  is  the  value  of  7  if  the  unit  is  one?  1  ten?  1 
hundred?  1  thousand? 

What  is  the  unit  of  4  in  43?  Ans.  1  ten? 

What  is  the  local  value  of  4?  Ans.  4  tens  or  40. 

In  3725  give  the  unit  and  local  value  of  each  figure. 

29.  TABLE  OF  TENS. 

10  Units                  make  1  Ten 10 

10  Tens                       "        1  Hundred     ....  100 

10  Hundreds             "       1  Thousand    ....  1000 

10  Thousands            "        1  Ten-thousand .     .     .  10000 

10  Ten-thousands      "        1  Hundred-thousand  .  100000 

30.  TABLE  OF  THOUSANDS. 

1000  Units  make  1  Thousand   .     .  1,000 

1000  Thousands         "       1  Million  .     .     .          1,000,000 
1000  Millions  "       1  Billion    .     .     .    1,000,000,000 

31.  TABLE  OF  PLACES  OR  ORDERS. 
A  figure  in  the 

1st   place  denotes  Ones  called  units  of  the  1st  order. 

2d       "  "        Tens  "         "      2d       " 

3d       "  "        Hundreds  "         "      3d       " 

4th     "  "        Thousands  "         "      4th      " 

5th     "  "        Ten-thousands          "         "      5th     " 

6th     "  "       Hundred-thousands "         "      6th     " 


Notation  and  Numeration.  49 

32.  TABLE  OF  PERIODS. 

The  1st    period  represents  Ones. 

2d         "  "  Thousands. 

3d  "  Millions. 

4th  "  Billions. 

5th       "  "  Trillions. 

33.  NUMERATION  TABLE. 

•          •          •          •          • 
to        to       to       to       co 

o       o>       u       o       5 


fl  °°  <§   s  °°  <£   c  ^  ^ 

WT^ 


^—     ,  i-  j^      /  £•      O  G      £  C 

WHO     WHO     WHO     WHO    WHO 

r-j      ;-J      r^  ^H      ,-i      r~*  r*      £*      r*  ,          .         _• 

Places  of  units i2^£2       S^32      ^^^      ^^-^      eooii-i 

Figures 360,504,725,913,876. 

Periods  I  Numbers Fifth.       Fourth.       Third.      Second.        First. 

\  Names Trillions.   Billions.   Millions.  Thousands.   Ones. 

EXERCISES  IN  NUMERATION. 

34.  i.  Read  the  number  represented  by  37000401064. 

OPERATION. — Separate  the  number  into  periods,  thus :  37,000, 
401,064.  The  4th  period  is  billions,  the  3d  is  millions,  the  2d 
is  thousands,  the  1st  is  units ;  hence,  the  number  is  37  billion 
0  million  401  thousand  64  units,  or  thirty-seven  billion  four 
hundred  one  thousand  sixty-four. 

RULE. — I.  Begin  at  the  right  and  separate  the  number 
into  periods. 

II.  Then  begin  at  the  left  and  read  each  successive  period 
as  if  it  stood  alone,  giving  each  its  name  except  the  period 
of  units. 

N.  I.-4. 


50 


Intermediate  Arithmetic. 


In  this  manner  read 


2. 

43564. 

8. 

3. 

75031. 

9. 

4. 

132140. 

10. 

5. 

5720307. 

11. 

6. 

4006009. 

12. 

7. 

7205806. 

13. 

54311237. 

801603709. 

4321780651. 

123456789. 

123456654321. 

230405060708090. 


EXERCISES  IN  NOTATION. 


35.  1.  Express  in  figures  fifty-three  billion  sixty-five 
million  three  hundred  seven. 

OPERATION. — Since  billions,  the  highest,  number  named,  occupy 
the  fourth  period,  there  will  be  four  periods  in  the  number.  Now, 
beginning  at  the  left,  we  fill  each  of  the  periods  with  the  given 
numbers  of  billions,  millions,  thousands,  and  units,  as  if  each  stood 

alone,  and  obtain 

53,065,000,307. 

KULE  I. — Consider  from  the  greatest  number  named  the 
necessary  number  of  periods. 

II. — Begin  at  the  left  and  fill  each  of  the  successive  pe- 
riods as  if  it  stood  alone. 

NOTE. — There  must  be  three  figures  in  every  period,  except  the 
one  at  the  left,  which  may  have  one,  two,  or  three.  All  vacant 
orders  and  periods  must  be  filled  with  ciphers. 

In  this  manner  express  in  figures  : 

2.  Eighteen  thousand  five  hundred  thirty-six. 

3.  Thirty-two  thousand  eight. 

4.  Forty-seven  thousand  two  hundred. 

5.  Two  hundred  forty  thousand  five  hundred  one. 


Notation  and  Numeration.  51 

6.  Six  million  five  thousand   forty-seven. 

7.  Nine  million  twenty-three  thousand  thirty-one. 

8.  Twenty-nine   million   four  hundred  twelve  thou- 
sand five. 

9.  One  hundred  seven  million  eleven  thousand  one 
hundred  four. 

10.  Seven  hundred  thirty  million    six  hundred  nine 
thousand  three  hundred  ninety-two. 

11.  Thirteen    billion    thirteen  million  thirteen   thou- 
sand thirteen. 

12.  Two  hundred  forty-five  billion  one  hundred  seven 
million  fifty  nine  thousand  eight  hundred  seventy. 

NOTATION  OF  DOLLARS  AND  CENTS. 

36.  The  Sign   of  Dollars  is  8,  which   is   read,  dollars. 
Thus,  $12  is  read  12  dollars. 

37.  The   Sign  of  Cents   is   c,  or    cts.,   which    is   read, 

cents. 

Thus,  23c.,  or  23  cts.,  is  read  23  cents. 

38.  Dollars  and  cents  may  be  written  as  one  number 
by  placing  a  point  (.)  between  them. 

Thus,  25  dollars  34  cents,  is  written  $25.34. 

39.  Since   it   takes  100  cents  to  make  a  dollar,  cents 
always  occupy   two   places   at   the   right  of  the   point. 
Hence,  when   the   number  of  cents   is   less   than  10,  a 
cipher    must    he    written    between    it    and    the    point. 

Thus,  5c.  is  written  $.05  ;  and  3  dollars  8  cents  is  written  $3.08. 
Neither  the  sign  ($)  nor  the  point  (.)  should  be  omitted. 


52  Intermediate  Arithmetic. 

40.  Exercises   in    Numeration    of  Dollars   and  Cents. 
Read  : 


1.  85.35. 

2.  $7.40. 

3.  810.09. 

4.  $.06. 


5.  $17.13. 

6.  $33.07. 

7.  $  .34. 

8.  $1.01. 


9.  $3140.43. 

10.  $504.67. 

11.  $5008.03. 

12.  $.04. 


41.  Exercises  in  Notation  of  Dollars  and  Cents.     Ex- 
press in  figures  and  signs  : 

1.  Thirteen  dollars  fifteen  cents. 

2.  Forty  dollars  fifty  cents. 

3.  Forty-three  dollars  seven  cents. 

4.  One  hundred  dollars  twenty  cents. 

5.  Sixty  dollars  ten  cents. 

6.  Thirty-five  cents. 

7.  Nine  cents. 

8.  Eighty-four  dollars  six  cents. 

9.  Ninety-nine  dollars  twelve  cents. 
10.  Fifty-four  cents. 

42.  A  Scale  in  Arithmetic  is  the  relation  between  the 
successive  orders  of  units. 

In  the  Arabic  system  of  notation,  the  scale  is  ten  ;  that  is,  the 
value  of  the  unit  in  any  order  is  ten  times  as  great  as  the  unit  in 
the  next  lower  order ;  hence,  it  is  called  the  Decimal  Scale,  from 
the  Latin  word  decem,  meaning  ten. 

NOTATION  OF  OBJECTS. 

43.  In  this   work   objects   are   frequently  denoted   by 
the  first  letters  of  their  names.      Thus,  an  arm,  an  ap- 
ple, etc.,  is  denoted  by  a;  a  box,  a  bin,  a  boy,  etc.,  is 
denoted  by  b. 


Notation  and  Numeration.  53 

Conversely,  an  a  may  be  regarded  as  denoting  an  apple,  or  an 
ax,  etc.;  li  as  denoting  an  hour,  a  hand,  etc.  If  a  letter  appears 
more  than  once  in  the  same  example,  each  one  denotes  the  same 
thing.  Thus:  2  b  and  1  b  are  3  b,  may  be  read,  2  boys  and  1 
boy  are  3  boys ;  or  2  boxes  and  1  box  are  3  boxes. 

ABBREVIATIONS. 

44.  The  colon  (:)  is  employed  to  denote  " the  follow- 
ing•,"  or  "as  follows"  and   to  signify  that    the   term  or 
phrase   preceding   it   is   to  be   prefixed   to   each   of  the 
phrases  following  it.    Thus,  "from  5  take  :  3,  4;"  means 
"from  5  take  3;  from  5  take  4." 

QUESTIONS  FOR  REVIEW. 

45.  What  is:  1.  A  unit?   2.  A  number?    8.  The  unit  of  a  num- 
ber ?    4.  An  abstract  number  ?    5.  A  concrete  number  ?    6.  Nota- 
tion?    7.  Numeration?    8.  A  scale? 

Name:  1.  The  figures.     2.  The  digits. 

What  is  the  greatest  number  that  can  be  expressed  by:  1.  One 
figure?  2.  Two  figures?  3.  Three  figures?  4.  Four  figures? 

How  many  figures  in  :  1.  One  place?    2.  One  period? 

How  are  places  and  periods  numbered  ? 

Repeat  the  table  of :  1.  Tens.  2.  Thousands.  3.  Places.  4. 
Periods. 

What  is  the  rule  for:  1.  Reading  numbers?  2.  Writing  num- 
bers? 

What  stands  for:  1.  Dollars?  2.  Cents?  How  arc  dollars  and 
cents  written  as  one  number? 

In  this  book:  What  often  stands  for  an  object?  What  does  a 
colon  ( : )  denote? 


A 


DDITION. 


INDUCTIVE  EXERCISES. 

48.     1.  Count  7.     Am.  1,  2,  3,  4,  5,  6,  7. 

2.  Count  $7.  ^m-.  81,  $2,  $3,  $4, 

3.  Iii  counting,  what  are  the  next  3  numbers  above 
$4?  Ans.  $5,  $6,  $7. 

4.  If  I  count  $4  and  then  count  S3  more,  how  much 
will  I  have  counted  in  all?  Ans.  $7. 

5.  In  counting,  what  are  the  next  4  numbers  above  6 
hats?  Ans.  7  hats,  8  hats,  9  hats,  10  hats. 

6.  If  I   count  6  hats  and   then   count  4  more,  how 
many  hats  will  I  have  counted  in  all?         Ans.  10  hats. 


In  counting,  what  is  the : 

7.  1st  number  above  8? 

8.  2d   number  above  9  ? 

9.  3d   number  above  $5  ? 

10.  4th  number  above  8  hours? 

11.  5th  number  above  4  pecks? 


How  many  are : 

7.  8  and  1  ? 

8.  9  and  2? 

9.  85  and  $3  ? 

10.  8  hours  and  4  hours  ? 

11.  4  pecks  and  5  pecks? 


12.  James  makes  7  steps  and  then  4  steps  more;  how 
many  steps  does  he  make  in  all? 

13.  If  you  have  5  apples  and  I  give  you  3  more,  how 
many  apples  will  you  then  have? 

14.  Seven  peaches  are  on  one  tree  and  6  on  another 
tree ;  how  many  peaches  on  both  trees  ? 

15.  John   paid   8   cents   for    apples   and   4   cents    for 
pears;  how  much  did  he  spend? 

(54) 


Addition.  55 


How  many  are  : 

16.  $6  and  $1. 

17.  7  men  and  2  men? 

18.  5  pints  and  3  pints. 


How  many  are: 


16.  $1  added  to 

17.  2  men  added  to  7  men? 

18.  3  pints  added  to  5  pints? 


DEFINITIONS. 

47.  Like  Numbers  are  those  which  have  units  of  the 
same  kind,   as  7  and  13;  3  pounds  and  11  pounds;  6 
tens  and  20  tens. 

48.  Unlike  Numbers  are  those  which   have   units  of 
different  kinds,  as  5  yards  and  17  pints ;  3  days  and  25 
feet. 

49.  Addition  is  the  process  of  uniting  two  or  more  like 
numbers  into  one, 

50.  The  numbers  added  are  called  the  parts,  and  the 
number  obtained  by  adding,  the  sum  or  amount. 

Thus,  7  balls  and  3  balls,  when  united,  make  a  group  of  10 
balls.  Here  7  balls  and  3  balls  are  the  parts,  and  10  balls  the 
sum  or  amount. 

51.  A  Sign  is  a  symbol  used  to  indicate  an  operation 
or  relation. 

52.  The  Sign  of  Addition  is  -f-  >  which  is  read :  and  or 
plus ;  plus  means  more.     When  -}-  stands  between  two 
numbers,  it  indicates  that  they  are  to  be  added. 

53.  The  Sign  of  Equality  is  = ,  which  is  read :  are  or 
equals. 

54.  The   Sign   of  Interrogation   is   ? ,  which   is   read  : 
what  or  how  many.     It  signifies  that  the  answer  is  to  be 
found,  and  when  found,  belongs  in  the  place  occupied 
by  the  sign. 


56  Intermediate  Arithmetic. 

Thus,  9  +  5  =  14  is  read:   9  and  5  are  14,  or  9  plus  5  equals 
14.    Beginners  should  adopt  the  first  reading. 

Again,  8 -f  3  =  ?  is  read:  8  and  3  are  how  many. 

EXERCISES. 
55.  Read : 


1.  8  +5  —13. 

2.  9  -1-6  =15. 

3.  5  +4  =? 

4.  6c.  -f  4c.  =  lOc. 

56.  Express  by  signs 


5.  $7  +  $10  =  $17. 

6.  $6 +  $3   =? 

7.  7  men  -f  4  men  =  11  men. 

8.  5  pins  +  6  pins  =  ? 


1.  3  and  5  are  8. 

2.  2  and  7  arc  what  ? 

3.  3 -f- 4  equals  what  ? 

4.  3c.  and  lOc.  are  13c. 


5 .  6  balls  added  to  3  balls  are  9  balls. 

6.  How  many  are  3  and  12? 

7.  The  sum  of  6  and  2  is  8. 

8.  $4  and  $8  are  how  many  ? 


57.  PRINCIPLE. — Only  like  numbers  can  be  added. 

58.  The  Complemental  Parts  of  a  number  are  the  num- 
bers whose  sum  equals  that  number. 

Thus:   the    complemental    parts  of   7  are  3  and  4,  or  1,  2,  3 
and  1. 

59.  By  Addition  we  find  a  number  when  its  comple- 
mental parts  are  given. 

Complemental  will  frequently  be  denoted  by  the  letter  c. 

Thus,  the  c  parts  of  8  are  5  and  3,  or  6  and  2. 

SUGGESTION  TO  TEACHERS. — In  all  the  examples  in  Addition,  the 
pupil  should  be  required  to  point  out  the  c  parts  and  the  ivliole. 

60.  The   following   table  should  be   thoroughly  com- 
mitted to  memory  : 


Addition. 


57 


ADDITION  TABLE. 


1. 

2. 

3. 

4. 

5. 

1  +  1=  2 

2+1=  3 

3+1=  4 

4+1=  5 

5+1=  6 

1-1-2=  3 

2+2=  4 

3  +  2=  5 

4+2=  6 

5+2=  7 

1+3=  4 

2+3=  5 

3+3=  6 

4+3=  7 

5+3=  8 

l-f4=  5 

2+4=  6 

3+4=  7 

4+4=  8 

5+4=  9 

14-5=  6 

2+5=  7 

3+5=  8 

4+5=  9 

5+5=10 

1+6=  7 

2+6=  8 

3+6=  9 

4+6=10 

5+6=11 

1+7=  8 

2+7=  9 

3+7=10 

4+7=11 

5+7=12 

1+8=  9 

2  +  8=10 

3+8=11 

4+8=12 

5+8=13 

1+9=10 

2+9=11 

3+9=12 

4+9=13 

5+9=14 

6. 

7. 

8. 

9. 

10. 

6  +  1—  7 

7+1=  8 

8+1=  9 

9+1—10 

10+1  —  11 

W         |          J.                                f 

J-  Vy     l     J.  l_  J. 

6+2=  8 

7+2=  9 

8+2=10 

9+2=11 

10+2=12 

6+3=  9 

7+3=10 

8+3=11 

9+3=12 

10+3=13 

6+4—10 

7-4-4—11 

8+4=12 

9+4=13 

10+4=14 

\-r          |             1    ^^^^  J_  \_T 

9         |        I  -L  J. 

6+5=11 

7+5=12 

8+5=13 

9+5=14 

10+5=15 

6+6=12 

7+6=13 

8+6=14 

9+6=15 

10+6=16 

6+7=13 

7+7=14 

8+7=15 

9+7=16 

10+7=17 

6+8=14 

7+8=15 

8+8=16 

9+8=17 

10+8=18 

6+9=15 

7+9=16 

8+9=17 

9+9=18 

10+9=19 

NOTE. — Pupils  should  read  the  tables  thus:  1  and  1  are  2, 
1  and  2  are  3,  2  and  1  are  3,  1  and  4  are  5,  4  and  1  are  5,  etc. 
The  table  is  expressed  in  signs  to  familiarize  the  pupil  with  their 
use  and  meaning. 

DRILL  EXERCISES. 


61.  The  following,  read   vertically,  are   the   two  com- 
pie  mental  parts  of  the  number  written  under  them : 


58 


Intermedia fe  Arithmetic. 


2 


32  43  543  654  7654  8765  98765 
12  12  123  123  1234  1234  12345 


6 


8 


9 


9      9 


98 

78 


18    17      16       15 


98      987      987      9876 
67      567      4j>_6      3456 

13 


14 


12 


10 

9876 
2345 

11 


SUGGESTIONS  TO  TEACHERS. — These  parts  and  their  sum  should 
be  thoroughly  committed  to  memory,  so  that  they  will  be  recog- 
nized at  a  glance.  For  this 
purpose  the  following  is 
recommended :  Procure  two 
boards  of  convenient  di- 
mensions and  dress  them 
smoothly,  so  that  one  will 
slide  freely  on  the  other. 
Mark,  number,  and  place 
j  them  as  shown  in  the  dia- 
gram, and  require  the  class 
in  daily  drill,  to  name  the 
sum  of  the  opposite  num- 
bers. Then  shift  the  posi- 
tion of  the  upper  board, 
and  proceed  as  before, 
until  all  the  combinations 
have  been  reached.  These 
boards,  especially  with  the 
addition  of  two  or  three 
similar  ones,  will  afford  abundant  drill  exercises  in  Addition, 
Subtraction,  and  Multiplication.  We  shall  subsequently  refer  to 
them  as  the  Combination  Boards.  See  Art.  104. 


COMBINATION  BOARDS. 


MENTAL  EXERCISES. 

62.   i.  Frank  is  10  years  old,  and  his  sister  is  3  years 
older;  how  old  is  his  sister? 


Addition.  59 

2.  John  has  6  pears,  and  William  has  5   more  than 
John  ;  how  many  pears  has  William  ? 

3.  Susan    has   7    roses   and    Lucy    has    3    roses;    how 
many  roses  have  both  girls? 

4.  Thomas  killed  6  squirrels  on  Monday,  and  8  squir- 
rels on  Friday ;  how  many  squirrels  did  he  kill  on  both 
days? 

5.  7  rabbits  are  in  the  garden,  and  4  rabbits  are  in 
the  yard ;  how  many  rabbits  in  all  ? 

6.  1  hen  has  8  chicks,  and  another  hen  has  5  chicks; 
how  many  chicks  have  both  hens? 

7.  A  boy  worked  9  hours  on  Tuesday,  and   6  hours 
on  Wednesday ;  how  many  hours  did  he  work  on  both 
days? 

8.  John   caught  5   fishes,  and   Ben  caught   8   fishes; 
how  many  were  caught  by  both  boys? 

9.  Moses  is  9  years  old  now;  how  old  will  he  be  7 
years  from  now  ? 

10.  Emma  has  8  tulips,  and  Ann  has  8  tulips  more 
than  Emma ;  how  many  tulips  has  Ann  ? 

11.  A  man  traveled  6  miles,  and  then  went  3  miles 
further;  how  far  did  he  travel  in  all? 

12.  Henry  spent  10  cents  for  oranges,  and  6  cents  for 
apples ;  how  much  did  he  spend  in  all  ? 

13.  How  many  are  2  and  2?     3  and  4?     4  and  5? 
6  and  6? 

14.  7  and  7?     8  and  8?     9  and  9?    5  and  6?     7  and 
3?     8  and  4? 

15.  9  and  2?    5  and  7?     6  and  8?     7  and  9?     2  and 
5?     1  and  8? 

16.  6  and  4?    5  and  5?     8  and  2?    5  and  3?     9  and 
9?    6  and  7? 

17.  What  is  the   number  whose  complemental   parts 
are  2  and  3?     3  and  4?    4  and  5?    5  and  6?     9  and 


60  Intermediate  Arithmetic. 

4?     7  and  8?     1,  3,  and  4?     2,  3,  and  7?     3,  5,  and  8? 
1,  2,  3,  and  4? 

18.  How  many  are  5  and  6?     15  and  6?     25  and  6? 
35  and  6?     45  and  6?     55  and  6?     65  and  6?     75  and 
6?     85  and  6?     95  and  6? 

19.  How  many  are  8  and  7?     18  and  7?     28  and  7? 

38  and  7  ?     48  and  7  ?    58  and  7  ?     68  and  7  ?     78  and 
7?     88  and  7?     98  and  7? 

20.  How  many  are  9  and  8?     19  and  8?     29  and  8? 

39  and  8?  etc.,  to  107. 

21.  How  many  are  4  and  9?     14  and  9?     24  and  9? 
etc.,  to  103. 

22.  How  many  are  7  and  5?     17  and  5?     27  and  5? 
etc.,  to  102. 

23.  How  many  are  4  and   10?     14  and   10?     24  and 
10?  etc.,  to  104." 

24.  How  many  are  4  and  5?     5  more  than  that?     5 
more    than    that?     5    more    than    that?     5   more   than 
that? 

25.  How  many  are  7  and  3?     3  more  than  that?     3 
more  than  that?     3  more  than  that?  etc.,  to  22. 

26.  How  many  are  8  and  4?     4  more  than  that?  etc., 
to  32. 

27.  How  many  are  9  a's  and  6  a's?     6  a's  more  than 
that?  etc.,  to  33  a's. 

28.  How  many  are  6  c's  and  7  c's?     7  c's  more  than 
that?  etc.,  to  41  c's. 

29.  How   many   are   2   ii's   and   8   ii's?     8   ii's   more 
than  that?  etc.,  to  32  n's. 

30.  How  many  are  3  cows  and  6  cows?     7  tens  and 
4  tens?     9  fives  and  3  fives?    5  thirds  and  2  thirds? 
8  tenths  and  4  tenths? 


Addition.  61 

EXERCISES  IN  MAKING  PROBLEMS. 

63.  i.  Make  a  problem  of  5  b  +  7  b  =  ? 

Ans.  5  bats  and  7  bats  are  how  many  ? 
Or,  Mary  has  5  beads  and  Susan  has  7  beads;  how 
many  beads  have  both  girls  ? 

2.    Make   a   problem   of  6  111  +  8  ill  : 

Ans.  6  men  and  8  men  are  how  many  ? 
Or,  there  are  6  marbles  in  one  pile,  and  8  marbles  in 
another  pile;  how  many  marbles  in  both  piles? 

Make  problems  of  the  following,  and  then  give  the 
answers : 


6.  36a  +  5a=  =? 

7.  54d  +  9d  =  ? 

8.  lOp  +  6p  +  5  p 


3.  7  c  +  8  c=? 

4.  6  f  +9  f  =? 

5.  10  h  +  5  h  =  ? 


SLATE  EXERCISES. 

64.  Copy  and  add  the    following,  placing  the  answer 
under  the  line  : 

4        9        5        7        8        4        7       10      34      46      57 
76475989389 


3 

5 

3 

5 

6 

5 

3 

8 

42 

34 

16 

2 

4 

0 

4 

3 

2 

5 

9 

2 

5 

4 

4 

7 

8 

9 

7 

2 

4 

7 

7 

3 

6 

5 

3 

1 

2 

3 

4 

3 

4 

3 

4 

5 

2 

7 

9 

7 

5 

6 

7 

8 

1 

6 

7 

1 

2 

2 

5 

5 

6 

3 

8 

9 

7 

8 

4 

5 

8 

7 

5 

6 

7 

4 

8 

10 

8 

62  Intermediate  Arithmetic. 


7 

3 

9 

6 

5 

7 

4 

3 

5 

8 

5 

8 

3 

4 

8 

9 

2 

3 

6 

9 

7 

5 

9 

5 

3 

7 

9 

5 

8 

6 

9 

10 

3 

8 

10 

5 

4 

12 

15 

17 

27 

30 

20 

36 

17 

The  following  exercises  may  be  performed,  first  on 
the  slate,  and  then  mentally  : 

MENTAL  EXERCISES. 

65.  i.  Name  every  10th  number  from:  0  to  100:  5  to 
105;  2  to  102;  7  to  107;  9  to  109;  3  to  103:  8  to  108; 
4  to  104;  1  to  101;  6  to  106. 

2.  Count  8  on  to:  7;  15;  21;  34;  46;  52;  68;  72;  89; 
97. 

3.  What  is  the  sum  of:  23  and  7?  37  and  5  ?  48  and 
3  ?    59  and  4  ?     66  and  8  ?    72  and  9  ?    84  and  7  ?    99 
and  3?  45  and  5  ?   76  and  7?   93  and  5  ?    87  arid  8?    63 
and  8? 

4.  16  +  7  =  =?   35  +  6=?  43  +  8=?   74  +  3  =  ?  98  + 
7  =  ?  57  +  9=? 

5.  To  every  10th  number  from  5  to  95  add:  7;  3;  8. 


Add  or  count : 

6.  By  2's  from  0  to  18. 

7.  By  3's  from  0  to  30. 

8.  By  4's  from  0  to  40. 

9.  By  5's  from  0  to  50. 

Name : 


10.  By  6's  from  0  to  60. 

11.  By  7's  from  0  to  70. 

12.  By  8's  from  0  'to  80. 

13.  By  9's  from  0  to  90. 


14.  Every  3d    number  from  2  to     32. 

15.  Every  4th  number  from  3  to     39. 


Addition. 


63 


16.  Every  5th  number  from  4  to  104. 

17.  Every  6th  number  from  5  to     65. 

18.  Every  7th  number  from  6  to     76. 

19.  Every  8th  number  from  7  to     87. 

20.  Every  9th  number  from  8  to     71. 

21.  What  pairs  of  digits,  when  added,  will  make:  4? 
5?  6?   7?    8?    9?    10?    11?    12?    13?    14?    15?    16? 


Find  the  sum  of: 

22.  5,  7,  9,  3. 

23.  2,  4,  6,  8. 

24.  7,  1,  9,  3,  5. 

25.  10,  6,  8,  4,  7. 

26.  $7,  $8,  $9. 

27.  6c,  7c,  3c,  9c. 

28.  5  hats,  4  hats,  7  hats. 

29.  16  m,  10  m,  9  m,  7  m. 

38.  45  +  9  +  7- 

39.  How  many 


30.  $50,  $10,  $9,  $7,  $3, 

31.  6  +  8  +  3=? 

32.  9  +  5+  6  =  ? 

33.  10  +  6  +  7  +  3  =  ? 

34.  23  +  7  +  6  +  5  =  ? 

35.  7  +  3+9  =  ? 

36.  9  +  7  +  8  =  ? 

37.  34  +  10+6  +  5  =  ? 

Q      —  9 

are  35  and  20? 


OPERATION. — 3  tens  and  2  tens  =  5  tens  or  50.     50  +  5  --  55. 

40.  There  are  45  boys  and  50  girls  in  school.      How 
many  pupils  in  all? 

41.  If  Charles   reads  78  pages  one  day  and   60  pages 
the  next  day,  how   many  pages   will  he   read   in   both 
days? 

42.  A  drover  bought  67  sheep  from  one  man  and  30 
from  another,  how  many  sheep  did  he  buy  of  both  ? 

43.  How  many  are  63  and  20?     78  and  40?    37  and 
70?     45  and  30?    97  and  40?    50  and  23?    70  and  85? 
90  and  44  ?    80  and  56  ?    60  and  84  ? 

44.  How  many  are  49  and  37? 


OPERATION.— 49  +  30+7  =  79 +  7  =  86. 


64  Intermediate  Arithmetic. 

45.  John  gave  25  cents  for  a  slate  and  42  cents  for  a 
book;  what  did  both  cost? 

46.  A  pole  is  43  feet  in  the  air,  19  feet  in  the  earth, 
and  .18  feet  in  the  water.     How  long  is  the  pole? 

47.  A   lad,  having   spent  43   cents,  finds   he   has   59 
cents  left.     How  much  had  he  at  first? 

48.  How  many  are  43  and  22?    64  and  53?    47  and 
35  ?    84  and  21  ?     74  and  56?     73  and  91  ?    53  and  41  ? 
75  and  92  ?    39  and  24  ?   43  and  87  ? 

49.  How  many  are  2  tens  and  12  ones? 

OPERATION. — 12  ones=l  ten  2  ones.  2  tens  and  1  ten  2  ones  are 
3  tens  2  ones  ==  32. 

50.  How  many  are  3  tens  and  25  ones?     Ans.  5  tens 
5  ones  =  55. 

51.  How  many  are:  4  tens  and  17  ones?     6  tens  and 
34  ones?    8  tens  and  73  ones? 

WRITTEN  EXERCISES. 

66.   1.  Find  the  sum  of  4864,  785  and  693.      OPERATION. 

EXPLANATION. — Write  the  numbers  or  parts,  so 
that  units  of  the  same  order  stand  in  the  same  col-  '  °5 

umn — ones  under  ones,  tens   under   tens  etc. ;   and 
draw  a  line  beneath  them.  6342 

Adding,  from  the  bottom,  the  column  on  the  right,  we  get  12 
(=1  ten  and  2  ones) ;  write  the  two  below  the  line  for  the  ones  of 
the  required  sum.  Adding  the  1  ten  with  the  tens  of  the  given 
parts,  which  is  the  next  column,  we  get  24  (=2  hundreds  and  4 
tens)  ;  write  the  4  below  the  line  for  the  tens  of  the  required  sum. 
Adding  the  2  hundreds  to  the  hundreds  of  the  given  parts,  which 
are  the  numbers  of  the  3d  column,  we  get  23  (=2  thousands  and  3 
hundreds)  ;  write  the  three  in  the  place  of  hundreds  in  the  sum, 
and  carry  the  2  to  the  next  column  of  thousands,  which,  added 
to  4,  gives  6  thousands.  Hence  the  sum  is  6342. 

To  prove  the  work  we  begin  at  the  top  and  add  down. 


Addition.  65 

2.  45364  -|-  8965  +  786  +  9374  -f  47  =  ?  OPERATION. 

45364 

67.  In  adding  it  is  best  to  use  only  the  follow-  8965 

ing  wording:  7,  11,  17,  22,  10  (emphasize  6,  and  786 

write  it  down  while  pronouncing  it),  carry  2 ;  6,  9374 

13,  21,  27,  33,  carry  3  ;  6,  13,  22,  25,  carry  2  ;  11,  1<),  47 

24,  carry  2  ;  6.  "6453G 

From  the  preceding  examples  we  derive  the  following 

RULE. — I.  Write  the  parts  so  that  like  orders  of  units 
shall  stand  under  each  other. 

II.  Begin  at  the  right,  add  each  column  separately,  write 
the  units7  figure  of  the  sum  under  the  column  added,  and 
carry  the  tens,  if  any,  to  the  next  column. 

PROOF. — Perform  the  addition  in  the  reverse  direction, 
from  top  to  bottom,  and  if  the  results  agree  the  work  is 
probably  correct. 

In  this  manner  add  and  prove : 


(3) 

87 

49 

64 

(6) 

37 

(7) 

65 

(8) 

48 

Ans. 

96 
183 

70 

89 

98 

34 

84 

(9) 

95 

(10) 

21 

(11) 
52 

(12) 

78 

(13) 

80 

(14) 

67 

36 

54 

76 

89 

63 

27 

48 

79 

81 

67 

94 

57 

Ans.  179 

(15) 

939 
827 
705 
693 

Ans.  3164 

N.  I.— 5. 


(16) 

(17) 

(18) 

(19) 

(20) 

818 

729 

60S 

590 

486 

706 

695 

587 

405 

705 

694 

581 

446 

967 

628 

582 

434 

993 

879 

530 

66 


Intermediate  Arithmetic. 


(21) 

(22) 

(23) 

(24) 

(25) 

38025 

75631 

9998 

642 

43384 

9467 

8467 

74635 

9753 

965 

7098 

983 

672 

85671 

8741 

Ans.  54590 

68.  Table  of  distances  on  the  Mississippi  River,  com- 
piled in  integers  of  miles,  from  the  surveys  of  the  Mis- 
sissippi River  Commission : 


Jetties  to  New  Orleans  .     .  96 

N.  O.  to  Donaldsville,  La.    .  78 

Donald,  to  Plaquemine,  La.  32 

Plaq.  to  Baton  Rouge,  La.    .  20 

B.  R.  to  Bayou  Sara,  La.      .  34 

B.  S.  to  Mouth  Red  R.,  La.  35 

Mouth  R.R.  to  Natchez,  Miss.  64 

Natchez  to  St.  Joseph,  La.  .  52 

St.  J.  to  Vicksburg,  Miss.     .  49 


Vicks.  to  La.  State  Line  .     .  47 

La.  S.  L.  to  Greenville,  Miss.  44 
Green,  to  Arkansas  City,  Ark.  40 

A.  C.  to  mouth  Ark.  R.,  Ark.  37 

Mou.  Ark.  R.  to  Helena,  Ark.  95 

Helena  to  Memphis,  Tenn.  76 

Memp.  to  Fort  Pillow,  Tenn.  58 

Fort  P.  to  Columbus,  Ky.    .  151 

Columbus  to  Cairo,  111.    .     .  21 


How  far  is  it  from  New  Orleans  by  river : 

26.  To  Baton  Rouge?  Ans.  130  miles. 

27.  To  Natchez?  Ans.  263      " 

28.  To  Vicksburg?  Ans.  364      " 

29.  To  Memphis?  Ans.  733      " 

30.  How  far  is  it  from  the  mouth  of  Red  River  to  the 
mouth  of  Arkansas  River?     To  the  Jetties? 

Ans.  363  miles ;  295  miles. 

31.  How  far  will  a  man  travel  who  takes  a  boat  at 
Baton  Rouge,  La.,  and  goes  to  Helena,  Ark.  Ans.  ? 

32.  How    far    is    it    from    the    Jetties    to    Memphis? 
Cairo?  Ans.  ? 

33.  A  farmer  sold  4  bales  of  cotton;  the  first  weighed 
463    pounds,    the    second    458    pounds,    the    third    417 
pounds,  and  the  fourth  513  pounds;  what  was  the  whole 
weight?  Ans.  1851  pounds. 


Addition.  67 

34.  An   army   is   composed   of    34379  infantry,    8625 
cavalry,  and  1792  artillery-men ;  how  many  men  in  the 
army?  Ans.  44796  men. 

35.  A    butcher   bought   6   oxen   which   weighed    1345 
pounds,   1623    pounds,  978   pounds,    1174    pounds,   819 
pounds,  and  1796  pounds  ;  what  was  the  total  weight  ? 

Ans.  ? 
Find  the  sum  of: 

36.  2564,  34875,  16374,  985,  76.         Ans.  54874. 

37.  14200  yards,  672  yards,  1265  yards,  3789  yards. 

Ans.  19926  yards. 

38.  340  acres,  281  acres,  57  acres,  426  acres,  5  acres. 

Ans.  1109  acres. 

39.  25  days,  460  days,  191  days,  763  days,  1084  days. 

An*.  ? 

NOTE. — When  numbers  of  dollars  and  cents  are  to  be  added, 
they  must  be  written  so  that  the  points  stand  under  each  other. 

40.  $36.27  +  85.96  +  $1208.  -f  $120.40  +  $75.00  +  $.94 
=  ?  Ans.  $1446.57. 

41.  $50.04  +  $7.80  +  $102.10  +  $15.08  +  $208.00  + 
$3.43  =  ?  Ans.  $386.45. 

42.  $304.00  +  $75.75  -f  $12.05  +  $27.54  +  85.81  + 
$63.02  =  ?  Am.  ? 

43.  What  is  the  amount  of  $45.63,  $3.68,  $37.45,  $93.07, 
$2.84,  $175.50,  and  $430.12?  Ans.  $788.29. 

44.  Mr.  Hicks  owes  Mr.  Johnson  $130.50,  Mr.  Jackson 
$475.12,  Mr.  Turner  $980,  and  Mr.  Wafer  $17.64;   how 
much  does  he  owe  them  all?  Ans.  $1603.26. 

45.  A   farmer  sold   a  horse  for  $109.50,  four  bales  of 
cotton   for  $197.85,  three  hogsheads  of  sugar  for  $239, 
and  a  load  of  corn  for  $13.25;  what  did  he  receive  for 
all?  Ans.   ? 

46.  North    America   contains   8,593,000  square   miles, 


68 


Intermediate  Arithmetic. 


South  America  7,362,000  square  miles,  Europe  3,825,000 
square  miles,  Asia  17,300,000  square  miles,  and  Africa 
11,557,000  square  miles;  how  many  square  miles  are  in 
these  five  countries?  Ans.  48,637,000  square  miles. 

69.  When  the  parts  are  equal,  the  whole  or  sum  may 
be  indicated  by  inclosing  one  of  the  equal  parts  in  a 
parenthesis,  and  writing  the  number  of  parts  before  it. 

Thus,  5  (7)  indicates  5  sevens,  or  7-|-7-|-7-f7-{-7  =  35. 


EXERCISES. 


i.  4  (357)=? 


OPERATION. 

357 
357 
357 
357 


1428 

6(8435.10)      =?  Ans.  ? 
7  (374  hogs)   =  ?  Ans.  ? 
4(25681bs.)    =?  Ans.  ? 
10  (504  cents)  =  ?  Am.  ? 


2.  5(645)    =?  Ans.  3225.  6. 

3.  7(807)      =?  ,4m.  5649.  7. 

4.  3(2768)=?  Ans.  8304.  8. 

5.  9(215)    =?  Ans.  1935.  9. 

10.  A   father   gave  each   of  his   five  sons  $125;   how 
much  did  he  give  them  all?  Ans.  $625. 

11.  What  is  the  total  weight  of  3  bales  of  cotton  if 
each  bale  weighs  410  pounds  ?  Ans.  1230  pounds. 

12.  James  has  5  boxes,  and  in  each  box  there  are  175 
chestnuts ;  how  many  chestnuts  has  he  in  all  ?     Ans.  ? 

13.  What  will  6  horses  cost  if  each  horse  cost  $150? 

Ans.  ? 
70.  PARALLEL  PROBLEMS. 

NOTE. — In  these  and  in  subsequent  parallel  problems  the  men- 
tal, denoted  by  in,  involve  the  same  principles  as  the  succeed- 
ing written  problem  or  problems,  and  are  intended  for  two  pur- 
poses, viz : 


Addition.  69 

1°.  To  supply  the  place  of  mental  arithmetic ;  hence,  they 
should  be  solved  mentally  and  recited  orally. 

2°.  To  furnish  indirectly  an  explanation  of  the  principles  and 
terms  embraced  in  the  parallel  written  problems,  so  that  the 
pupil,  by  proper  diligence,  may  comprehend  and  solve  the  latter 
unaided. 

I."1  What  is  the  9th  number  above  12? 

2.    What  is  the  837th  number  above  968?     Ans.  1805. 

3.m  I  sold  a  hog  for  $13,  which  lacked  $5  of  being  as 
much  as  the  hog  cost  me ;  what  did  the  hog  cost  ? 

4.  By  selling  a  farm  for  $1305  I  lost  $960.50;  what 
did  the  farm  cost  me?  Ans.  $2265.50. 

5.m  James  spent  $8  for  pants,  $9  for  a  coat,  and  $5 
for  a  vest;  how  much  did  he  spend  in  all? 

6.  A  merchant  invests  $3275  in  a  house,  $9760.23  in 
merchandise,  $2987.53  in  improvements,  and  $1624.35 
in  clerks'  hire ;  what  is  his  total  investment  ? 

Ans.  $17647.11. 

7.m  John  gave  his  mother  12  pears,  his  father  7  pears, 
his  sister  6  pears,  and  had  5  pears  left ;  how  many  had 
he  at  first? 

8.  A  farmer  paid  $4875.60  for  a  farm,  $1782  for  stock, 
$2416.98  for  supplies,  and  had  $3026.05  left;  how  much 
money  had  he  at  first?  Ans.  $12100.63. 

9.™  William  paid  $7  for  a  hog,  $9  for  a  calf,  $5  for  a 
sheep,  and  sold  them  so  as  to  make  $6;  what  did  he 
receive  for  all? 

10.  A  speculator  bought  a  drove  of  horses  for 
$3764.15,  a  drove  of  cattle  for  $2017.55,  a  drove  of 
sheep  for  $620,  and  sold  them  at  a  profit  of  $784.85 ; 
what  did  he  receive  for  all?  Ans.  $7186.55. 

ll.m  James  has  3  marbles,  John  5  more  than  James, 
and  Moses  6  more  than  John ;  how  many  have  all  ? 

12.     Mr.  Taylor  has  735    sheep,  Mr.  Jackson  has  842 


70  Intermediate  Arithmetic. 

more  than  Mr.  Taylor,  and   Mr.  Hulse  634   more   than 
Mr.  Jackson ;  how  many  have  the  three  men  ? 

Ans.  4523  sheep. 

I3.m  Six  years  ago  Peter  was  9  years  old;  how  old 
will  he  be  8  years  from  now? 

14.  A  man  married  17  years  since,  at  which  time  he 
was  24  years  old;  how  old  will  he  be  22  years  hence? 

Ans.  63  years. 

15.™  Add  by  9's  from  0  to  45. 
16.    Add  or  count  by  278's  from  0  to  1390. 

Ans.  0,  278,  556,  etc. 

71.  QUESTIONS  FOR  REVIEW. 

What  are:  1.  Like  numbers?  2.  Unlike  numbers?  What  is 
Addition?  What  is  the  sign  of  Addition?  What  is  denoted  :  1. 
By  the  sign  =  ?  2.  By  the  sign  (?)  ?  What  are  the  complemental 
parts  of  a  number  ?  What  do  we  find  by  Addition  ?  What  stands 
for  complemental  ? 

What  is  the  :  1.  Principle  of  Addition  ?  2.  Rule  for  Addition? 
How  may  we  prove  Addition  ? 

What  is  meant  by  parallel  problems?  Ans.  Problems  which 
involve  the  same  principles. 


SUBTRACTION. 


INDUCTIVE  EXERCISES. 

72.  l.  Count  7  backward.  Ans.  7,  6,  5,  4,  3,  2,  1. 

2.  Count  $7  backward.    Ans.  $7,  $6,  $5,  84,  $3,  $2,  $1. 

3.  How  do  we  count  backward? 

4.  In   counting  backward,  what  are  the   next  three 
numbers  below  $7? 

5.  How  many  are  left  when  $1  is  taken  3  times  from 

$7? 

6.  In  counting  backward,  what  are  the  next  4  num- 
bers below  10  hats?  Ans.  9  h's,  8  h's,  7  li's,  6  h's. 

7.  How  many  are  left  when  4  hats  are  taken  from 
10  hats?  Ans.    6  hats. 


In  counting  backward,  what  is: 

8.  The  1st  number  below  9? 

9.  The  3d  number  below  12  ? 

10.  The  3d  number  below  $8? 

11.  The  4th  number  below  12  c? 


How  many  are: 

8.  9  less  1? 

9.  12  less  3? 

10.  $8  less  83? 

11.  12c'sless4c's? 


12.  If  I   make   7   marks  ///////,  and   rub  out  3  of 
them,  how  many  will  be  left? 

13.  John  had  12  balls  and  gave  James  5;  how  many 
balls  did  John  have  left? 

14.  Eight  peaches  are  on  a  tree,  if  5  are  taken  off,  how 
many  will  be  left? 

(71) 


72 


Intermediate  Arithmetic. 


How  many  are: 

15.  $7  less  $1? 

16.  9  men  less  2  men? 

17.  8  pints  less  3  pints? 


Subtract : 

15.  $1  from  $7. 

16.  2  men  from  9  men. 

17.  3  pints  from  8  pints. 


DEFINITIONS. 

73.  Subtraction  is  the  process  of  taking  from  a  number 
a  given  number  of  like  units. 

74.  The  number  to  be  diminished  is  called  the  Minu- 
end, the  number  by  which  it  is  diminished,  the  Subtra- 
hend, and  the  result  the  difference  or  remainder. 

Thus  :  $3  taken  from  $7  leaves  $4.  Here,  $7  is  the  minuend, 
$3  the  subtrahend,  and  $4  the  remainder. 

75.  The  sign  of  Subtraction  is  —  ,  which  is  read  :  less  or 
minus.     When  —  stands  between  two  numbers  it  indi- 
cates that  the  one  after  it  is  to  be  taken  from  the  one 
before  it. 

Thus  :  8  —  3  —  5  is  read  :  8  less  3  is  5,  or  8  minus  3  equals  5. 
Beginners  should  adopt  the  first  reading,  or  :  3  from  8  leaves  5. 

Again,  10  —  4  =  ?  is  read:  10  less  4  is  how  many?  or  4  from 
10  leaves  how  many  ? 


76.  Read: 


EXERCISES 


1.  9  in's  —  6  m's  =  3  m's. 

2.  7d's  — 4d's=3  d's. 

3.  12  boys --7  boys=? 

77.  Express  by  signs: 

1.  $14  minus  $5  is  $9. 

2.  $12  less  $7  is  $5. 

3.  7  plus  5  minus  3  is  9. 


4.  6  +  5  —  3  =  8. 

5.  7  +  8--6--4  =  6. 

6.  10  —  6  +  5—3  =  6. 

4.  16  less  7  is  how  many? 

5.  3  from  11  leaves  8. 

6.  6  from  14  leaves  8. 


Subtraction.  73 

In  each  of  the  preceding  examples  point  out  the  minuend, 
subtrahend,  and  remainder.  Thus,  in  Ex.  3,  7  plus  5  is  the 
minuend,  3  is  the  subtrahend,  and  9  the  remainder. 

78.  PRINCIPLE. — Only  like  numbers  can  be  subtracted. 


RELATION  OF  SUBTRACTION  TO  ADDITION. 

79.  4  and  3  are  how  many?     4  and  what  number  are 
7  ?     What  number  and  3  are  7  ? 

4  -f  ?  =  7  ?  Ans.  3  ;  because  7  —  4=3. 

?  + 3=7?  Ans.  4;  because  7  —  3  =  4. 

In  a  similar  manner  answer  the  following: 

9  +  ?=  12.          ?  +  7  =  15.     5  cents  +?  =  11  cents. 
8  +  ?  =  14.         7  +  3  =  10.     ?  + 6  pecks  =  9  pecks. 

PRINCIPLES. 

1°.  Subtraction  is  the  reverse  of  Addition. 

2°.  By  Subtraction  we  find  one  of  the  complemental 
parts  of  a  number,  when  the  number  and  the  other  part 
are  given. 

Thus :  if  6  is  one  of  the  c  parts  of  11,  the  other  part  is  11  —  6, 
or  5. 

SUGGESTIONS  TO  TEACHERS. — In  all  the  examples  and  problems 
in  Subtraction,  the  pupil  should  be  required  to  point  out  the  c 
parts  and  the  whole. 

Thus,  in  11  -  -  5  ==  6,  5  and  6  are  the  parts,  and  11  is  the 
whole.  Again,  in  the  problem :  James  had  $15,  but  lost  $8, 
how  much  had  he  left?  $8  and  $7  are  the  parts,  and  $15  the 
whole. 

80.  Since  Subtraction  is  the  reverse  of  Addition,  by 
reversing  the  table  of  the  latter,  we  get  the 


74 


Intermediate  Arithmetic. 


SUBTRACTION  TABLE. 


1 

2 

3 

4 

5 

1--1==0 

2  —  2  =  0 

3  —  3  =  0 

4-  -4  =  0 

5  —  5  =  0 

2--l==l 

3  —  2  =  1 

4  —  3  =  1 

5--4  =  l 

6  —  5  =  1 

3--l==2 

4  —  2  =  2 

5  —  3  =  2 

6-  -4  =  2 

7  —  5  =  2 

4  —  1  =  3 

5  —  2  =  3 

6  —  3  =  3 

7--4  =  3 

8  —  5  =  3 

5  —  1-4 

6  —  2  =  4 

7  —  3  =  4 

8--4  =  4 

9  —  5  =  4 

6  —  1  =  =  5 

7  —  2  =  5 

8  —  3  =  5 

9  --4  =  5 

10  —  5  =  5 

7--I==G 

8  —  2  =  6 

9-3  =  6 

10--4  =  6 

11  —  5  =  6 

8  —  1  =  7 

9  —  2  =  7 

10  —  3  =  7 

ll--4  =  7 

12  —  5  =  7 

9  —  1  =  8 

10  —  2  =  8 

11  —  3  =  8 

12  --4  =  8 

13  —  5  =  8 

10  —  1  =  9 

11  —  2  =  9 

12  —  3  =  9 

13--4  =  9 

14  —  5=9 

6 

7 

8 

9 

10 

6  —  6  =  0 

7—7  =  0 

8  —  8  =  0 

9  —  9  =  0 

10  —  10  =  0 

7  —  6  =  1 

8  —  7  =  1 

9  —  8  =  1 

10  —  9  =  1 

11-  -10  =  1 

8  —  6  =  2 

9  —  7  =  2 

10  —  8  =  2 

ll--9  =  2 

12  —  10  =  2 

9  —  6  =  3 

10  —  7  =  3 

11  —  8  =  3 

12  —  9  =  3 

13  —  10  =  3 

10—6  =  4 

11  —  7  =  4 

12  —  8  =  4 

13  —  9  =  4 

14  —  10  =  4 

ll--6  =  5 

12  —  7  =  5 

13  —  8  =  5 

14  —  9  =  5 

15  —  10  =  5 

12  —  6  =  6 

13  —  7  =  6 

14  —  8  =  6 

15  —  9  =  6 

16  —  10  =  6 

13  —  6=7 

14  —  7  =  7 

15  —  8  =  7 

16  —  9  =  7 

17--10  =  7 

14  —  6  =  8 

15  —  7  =  8 

16  —  8  =  8 

17  —  9  =  8 

18  --10  =  8 

15  —  6  =  9 

16  —  7==9 

17  —  8  =  9 

18  —  9  =  9 

19  --10  =  9 

NOTE. — Pupils  should  read  the  tables  thus :  1  from  1  leaves 
none,  1  from  2  leaves  1 ,  1  from  3  leaves  2,  etc.  The  table  is  ex- 
pressed in  signs  to  familiarize  the  pupil  with  their  use  and  mean- 


ing. 


DRILL  EXERCISES. 


81.  In  these  exercises  each  figure  is  to  be  subtracted 
from  the  number  that  stands  above  the  group.  The 
exercises  should  be  written  on  the  board,  and  used  in 
class  drill  daily,  until  every  pupil  can  call  all  the  re- 
sults instantly. 


Subtraction.  7"> 

2345  6  7  8 

1       12      123      1324      13542       153246      lfi:V)437 

18     17       16         15  14  13  12 

9   89   798   6897   58796   486597   3856479 

9        10         11 

18347562    183659742    63752849 

MENTAL  EXERCISES. 

82.   1.   Susan  is   14  years  old,  and   her  brother   is  5 
years  younger;  how  old  is  her  brother? 

2.  William  has  11   pears,  and  John   has  6   less  than 
William ;  how  many  pears  has  John  ? 

3.  Susan  and   Lucy  together  have   10  roses  ;    if  7  of 
them  are  Susan's,  how  many  has  Lucy? 

4.  Thomas  killed  8  sqirrels  on  Tuesday,  and   14  on 
Tuesday  and  Wednesday  together  ;    how  many  did  he 
kill  on  Wednesday  ? 

5.  There  are  13  rabbits  in  the  garden  and  yard;  if  6 
rabbits  are  in  the  yard,  how  many  are  in  the  garden  ? 

6.  Two  hens  have  13  chicks  together;  if  one  hen  has 
8  chicks,  how  many  chicks  has  the  other? 

7.  A  boy  worked  15  hours  in  two  days ;  if  he  worked 
6  hours  in  one  day,  how  many  did  he  work  the  other? 

8.  Ben  caught  17  fishes  and  Moses  9  fishes  ;  how  many 
more  fishes  did  Ben  catch  than  Moses? 

9.  7  years  from  now  Henry  will  be  15  years  old ;  how 
old  is  he  now  ? 

10.  Emma  has  14  tulips  and  has  7  tulips  more  than 
Ann ;  how  many  tulips  has  Ann  ? 

11.  A  man  traveled  9  miles;  how  far  would  he  have 
traveled  if  he  had  gone  5  miles  less? 


76  Intermediate  Arithmetic. 

12.  Harry  spent  16  cents  for  oranges  and  8  cents  less 
for  apples;  how  much  did  he  spend  for  apples? 

13.  How  many  is:  13  less  6?     17  less  8?     8  less  5? 

14.  10  less  2?     10  less  5?    9  less  8?    7  less  5?     16 
less  8? 

15.  16  less  7?     14  less  6?     12  less  7?     11  less  9?    10 
less  6? 

16.  10  less  3?     11  less  5?     18  less  9?     16  less  9?     14 
less  7? 

17.  If  7  is  one  of  the  C  parts  of  15,  what  is  the  other? 

18.  If  9  is  one  of  the  c  parts  of  13,  what  is  the  other? 

19.  How  many  is:  104  less  6?     94  less  6?     84  less  6? 
74  less  6  ?     64  less  6  ?    54  less  6  ?    44  less  6  ?    34  less  6  ? 
24  less  6  ?     14  less  6  ? 

20.  How  many  is  :  103  less  7  ?    93  less  7  ?    83  less  7  ? 
73  less  7  ?    63  less  7  ?    53  less  7  ?    43  less  7  ?    33  less  7  ? 
23  less  7?     13  less  7? 

21.  How  many  is:  105  less  9?     95  less  9?  etc.,  to  6. 

22.  How  many  is:  101  less  10?     91  less  10?  etc.,  to  1. 

23.  How   many   is:   53  less   8?     8  less  than  that?     8 
less   than   that?     8  less  than  that?     8  less  than  that? 

24.  How  many  is:  43  less  5?    5  less  than  that?  etc., 
to  3. 

25.  How  many  is:  65  less  9?     9  less  than  that?  etc., 
to  2? 

26.  How  many  is  33  diminished  by  7  four  times? 

27.  How  many  is  30  less  5,  less  5,  less  5,  less  6? 

28.  17  a's  — 9  a's  =  ?     25  b's  -  6  b's  =  ?     32  e's  — 
5  c's—?    40  n's  — 10  n's=? 

EXERCISES  IN  MAKING  PROBLEMS. 

83.  1.  Make  a  problem  of:  11  S--5  s  =  =  ? 

Ans.  5  spoons  from  11  spoons  leaves  how  many? 


Subtraction.  77 

Or,  James  had  11  strings,  but  gave  5  strings  to  his 
sister;  how  many  did  lie  have  left? 

2.  Make  a  problem  of:  13y--6y==?  Problem:  Mike 
is  13  years  old,  and  Henry  is  6  years  old ;  how  much 
older  is  Mike  than  Henry? 

Make  problems  of  the  following,  giving  the  answers 
to  each  : 


6.  23  a     -  6  a     --  ? 

7.  32  b  —  7b  =? 

8.  65  in— 9 m  =  =? 


3.  9  c  -  -2c  =? 

4.  11  r  -  -8r  =? 

5.  17  b  —  9b=? 

9.  7a-f6a--5a  —  ?     Ans.   If  from   the   sum   of  7 
apples  and  6  apples  I  take  5  apples,  how  many  apples 
will  be  left? 

10.  10  b  +  6  b  —  3  b    =  ? 

11.  12  a  +  9  c  —  6  c    =? 

12.  24  g;  +8  g  +3  g    —  6  g  =  ? 

13.  15  in  -f-  6  ni  -    4  in  -  -  5  in  =  =  ? 

MENTAL  EXERCISES. 

84.  l.  Name  every  tenth  number  from:  100  to  0: 
105  to  5  ;  102  to  2 ;  107  to  7 ;  109  to  9 ;  103  to  3 ;  108 
to  8 ;  104  to  4 ;  101  to  1 ;  106  to  6. 

2.  Take  8  from:  15;  23;  29;  42;  54;  61;  76;  80;  97; 
105. 

3.  What    is    the    difference    between   23    and    7?    42 
and  5?     51  and  3?     63  and  4?     74  and  8?     81  and  9? 
91  and  7  ?     102  and  3  ?    50  and  5  ?     83  and  7  ?     98  and 
5?     95  and  8? 

4.  23-  -7:=?    41  --6  =  =?    51-   -8==?    77  -  -  3  =  ? 
105  —  7  =  ?     66  —  9  =  ? 

5.  To   every  10th  number  from   105  to  5  subtract  7; 
3;  8. 


78  Intermediate  Arithmetic. 


Subtract : 

6.  By  2's  from  18  to  0. 

7.  IJy  8s  from  30  to  0. 

8.  IJy  4's  from  40  to  0. 

9.  IJy  5's  from  50  to  0. 

Name  : 


10.  By  6fs  from  60  to  0. 

11.  By  7's  from  70  to  0. 

12.  By  8's  from  80  to  0. 

13.  By  9's  from  90  to  0. 


14.  Every  3d    number  from     32  to  2. 

15.  Every  4th  number  from     39  to  3. 

16.  Every  5th  number  from  104  to  4. 

17.  Every  6th  number  from     65  to  5. 

18.  Every  7th  number  from     76  to  6. 

19.  Every  8th  number  from     87  to  7. 

20.  Every  9th  number  from     71  to  8. 
21.5  —  7  —  4  =  ?  25.  85    +88    —84  =  ? 

22.  9+6—5=?  26.  89   +  87    -f  S6—  $3  =  ? 


23.  7  +  8  +  6  —  4  =  ? 

<w  i .     t/     j      i  O  ^  —   -  • 


27.  10c.-     4c.  +  7c.—  2c.=  ? 

28.  40  —12    —8   —5  =? 


29.  A    man   bought  a  cow   for  830,  and  paid  all  but 
810;  how  much  did  he  pay? 

30.  Henry  had  50  cents,  but  he  paid  20  cents  for  some 
paper  ;  how  many  cents  had  he  left  ? 

31.  40  gallons  of  syrup  have  been  drawn  from  a  tank 
that  contained  90  gallons;  how  many  gallons  are  left? 

32.  James  has   caught  60  fishes  and  Moses  40;  how 
many  more  fishes  must  Moses  catch  to   have  as  many 
as  James? 

33.  Emma  has  80  beads  and  Susan  has  20  less  than 
Emma;  how  many  beads  has  Susan? 

34.  A  man  owing  870  gave  his  note  for  815,  and  paid 
the  balance  in  cash ;  how  much  cash  did  he  pay  ? 

35.  I  gave  a  horse   worth   880  for  a  cow  and  818  in 
money;  how  much  did  the  cow  cost  me? 


Subtraction.  79 

36.  William  had  42  oranges  and  gave  9  to  Mary  and 
7  tu  James;  how  many  orun '_:••>  had  In-  It/ft? 

37.  Edward    had   ?<5-">,   and  aft<  r  s}M-ndin<r  6 15  lost  all 
the  balance  except  £12;  how  much  did  he  lose? 

38.  From  04  take  27.  . 

OPERATION-  :  i; !  -  -  20  =  44,  44  —  7  ==  37. 


39.  From  75  take  36. 

40.  From  42  take  17. 

41.  From  s:',  take  65. 

42.  From  62  take  21. 


43.  54      28    =? 

44. 

45.  91-  -74==? 

46.  107  —  39  =  =  ? 


WRITTEN  EXERCISES. 

85.  CASE  I. — When  each  figure  of  the  subtrahend  is 
less  than  the  corresponding  figure  of  the  minuend. 

I.  From  695  subtract  324. 

OPERATION*. 

EXPLANATION. — 695  --  6  hunds.  9  tens  5  ones.  695 

324  =  3  hunds.  2  tens  4  ones.  •>•)  f 

O-£-t 

Subtracting  like  numbers,  we  have  3  hunds.  t>>. 

tens  1  one  ==371.  3/1 

Hence,  the 

RULE. — I.   Write  the  subtrahend  under  the  minuend,  pluc- 
iny  unit*  titiflt-r  unit*,  tens  under  ten*.  ef>\ 

II.  Begin  at  the  right,  subtract  each  figure  of  the  siib- 
trahend  from    the  figure  above   it.  and   write  the  difference 
below. 

From  : 


2.  375  take  124.    Ans.  251. 


3.  ns;  take  505.     Ans.  182. 

4.  9Js:J  take  823.    An*.  160. 


5.  841  take    31.     Ans.  810. 

6.  9324  take  113.  Ans.  0211. 

7.  7379  take  7163.  Ans.  216. 


8.  2:V.)4<;  take  1814.  Ans.  22132. 

9.  68438  take  ^25.  Ans.  68213. 

10.  7<H;:M  take  S420.  Ans.  71214. 


80  Intermediate  Arithmetic. 

11.  John  had   84   apples  and  gave  his   mother  51  of 
them;  how  many  had  he  left?  Ans.  33  apples. 

12.  A  farm  contains  649  acres  ;  if  332  acres  should  be 
sold,  how  many  acres  would  be  left?  Ans.   ? 

13.  Henry  gathered  4845  chestnuts  and  gave  2104  of 
them  to  his  sister;  how  many  had  he  left?     Ans.  2741. 

14.  A  farmer  has  679  sheep  in  one  field,  and  420  sheep 
in  another  field.     How  many  more  sheep  in  one  field 
than  the  other?  Ans.   ? 

15.  How  many  more  are  872  gallons  than  501  gallons? 

Ans.  371  gallons. 

16.  One  of  the  parts  of  666  is  234;    what  is  the  c 
part?  Ans.  432. 

86.  CASE  II. — When  any  figure  in  the  subtrahend  is 
greater  than  the  corresponding  figure  in  the  minuend. 

I.  From  764  take  481. 

EXPLANATION. — Beginning  at  the  right,  we  pro-      OPERATION. 
ceed  as  before,  and  say  1  from  4  leaves  3  ;  since  734 

8  tern  is  more  than  6  tens,  we  take  1  bund,  from  AQI 

the  7  bunds.,  add  it  to  the  6  tens,  making  16  tens. 

oqo 

Now,  8  tens  from  16  tens  leave  8  tens,  and  since  ^° 

we  took  1  from  7,  we  say  4  from  6  leaves  2. 

Hence, 

RULE. — I.  Proceed  as  far  as  possible  as  in  Case  I. 

II.  When  any  figure  of  the  subtrahend  is  greater  than  the 
one  above  it,  add  10  to  the  latter  and  subtract,  then  diminish 
by  1  the  units  of  the  next  higher  order  in  the  minuend. 

PROOF.  —  Add  the  remainder  and  the  subtrahend;  their 
sum  should  be  the  minuend. 

87.  Instead  of  diminishing  by  1  the  units  of  the  next 
higher  order  in  the  minuend,  it   is  more  convenient  in 
practice  to  increase  by  1  the  units  of  the  next  higher 


Subtraction. 


81 


order   in   the   subtrahend.      This   is   illustrated    in   the 
next  example. 

3.  From  95283  take  8365. 


Subtract  thus :  5  from  13  leaves  8  ;  carry  1  to 
6  makes  7,  7  from  8  leaves  1 ;  3  from  12  leaves  9 ; 
carry  1  to  8  makes  9,  9  from  15  leaves  6 ;  carry  1, 
1  from  9  leaves  8. 


OPERATION. 
95283 
8365 


Copy,  subtract,  and  prove  • 


(4) 

76 
_38_ 

Ans.  38 

(8) 

701 
637 


(5) 

137 
96 

Ans.  41 

(9) 

591 
265 


Ans.   ? 

(12) 
$520 
$361 


Ans.  $159 


Ana.    ? 

(13) 
$32.00 
$18.25 

Ans.    ? 


(6) 
4672 
3708 

Ans.  964 

(10) 
4703 
3928 

Ans.    ? 


86918 

(7) 

6875 
928 

Ans.  5947 

(ID 
32534 

18276 
Ans.  ? 


(14)  (15) 

1371  days     70000  pins. 

2859  days     36527  pins. 


Ans.  ? 


Ans.  ? 


Subtract : 

16.  99  from  125.  Ans.  26. 

17.  97  from  185.  Ans.  88. 


18.  961  from  1728.  Ans.  767. 

19.  965  from  2873.  Ans.  1908. 


20.  9283  from  11678.     Ans.  2395. 

21.  5398  from  83119.     Ans.    ? 

What  is  the  difference  between : 

22.  5873  dollars  and  7325  dollars?  Ans.  $1452. 

23.  158701  gallons  and  98731  gallons?  Ans.    ? 

24.  1158  sheep --736  sheep  =  ?  Ans.  422  sheep. 

25.  7003  rocks  — 2635  rocks  =  ?  Ans.  4368  rocks. 

N.  I.— G. 


82 


Intermedia te  A r ith  metic. 


How  many  more  are: 

26.  5000  cows  than  3001  cows?  Ans.  1999  cows. 

27.  2375  threes  than  1725  threes?          Ans.  650  threes. 

28.  83201  units  than  64736  units?     Ans.  18465  units. 

29.  61111  thirds  than  51115  thirds?    Ans.  9996  thirds. 
so.  7123  a  than  6234  a?  -  Ans.  889  a. 

88.  A  list  of  a  few  celebrated  mathematicians  ;  their 
nativities,  inventions,  and  the  times  of  their  births  and 
deaths. 


NAME. 

NATIVITY. 

INVENTIONS. 

BIRTH. 

DEATH 

Bowditcli 

American 

Navigation  Tables 

1773 

1838 

Briggs 

English 

Common  Logarithms 

1556 

1630 

Cardan 

Italian 

Solution  of  Cubic  Equat'ns 

1501 

1576 

Demoivre 

French 

Trigonometrical  Formulas 

1667 

1754 

Descartes 

French 

Analytical  Geometry 

1596 

1650 

Euclid 

Greek 

Geometry 

B.  C.  300 

Hamilton 

Irish 

Science  of  Quaternions 

1805 

1865 

Lagrange 

French 

Calculus  of  Variations 

1736 

1813 

Leibnitz 

German 

Dif.  and  Integral  Calculus 

1646 

1716 

Monge 

French 

Descriptive  Geometry 

1746 

1818 

Napier 

Scotch 

Logarithms 

1550 

1617 

Newton 

English 

Binomial  Theor.,  Calculus 

1642 

1727 

Sturm 

Swiss 

Position  of  Real  Roots 

1803 

1855 

Taylor 

English 

Cal.  of  Finite  Differences 

1685 

1731 

How  old  was : 

31.  Cardan  when  Napier  was  born?         Ans.  49  years. 

32.  Napier  when  Descartes  was  born?      Ans.  46  years. 

33.  Descartes  when  Newton  was  born?     Ans.  46  years. 

34.  Newton  when  Demoivre  was  born  ?    Ans.  25  years. 

35.  Demoivre  when  Lagrange  wras  bom?  Ans.  69  years. 

36.  Lagrange  when  Bowditch  was  born?  Ans.  37  years. 


Subtraction.  83 

37.  How  old  was  Bowditch  when  Hamilton  was  born? 

Ans.  32  years. 

38.  At  what  age  did  Bowditch  die?     Monge?     New- 
ton?    Taylor?     Briggs?     Leibnitz?     Sturm? 

39.  How   many   years   had   Napier  been   dead   when 
Newton  was  born? 

40.  A  man  had  $625  and   spent  $235.75;   how  many 
dollars  did  he  have  left? 

OPERATION. 

EXPLANATION. — We  write  the  amounts  so  that 
the  points  stand  under  each  other,  annex  two  O's 
to   the   minuend   to   supply  the  vacant  places  of          235.75 
cents,  and  subtract  as  in  simple  numbers.  389.25 

41.  A  man  bought  a  horse  for  $187  and  a  buggy  for 
$118.35;  how  much  more  did   the  horse  cost  than  the 
buggy?  Ans.  $68.65. 

42.  A  farmer  bought  a  wagon    for  8113,  and  gave  in 
exchange  a  cow  worth  $43.75,  and  the  balance  in  cash ; 
how  much  was  the  balance?  Ans.  $69.25. 

43.  Mount  Sorata,  a  peak  of  the  Andes,  is  21,286  feet 
high,  which  is  5,506  feet  higher  than  Mount  Blanc,  the 
highest  peak  of  the  Alps;  how  high  is  Mount   Blanc? 

Ans.   15780  feet. 

44.  In   1840   there   were   2428921    inhabitants   in   the 
State   of  New   York,  and    1724033    inhabitants   in    the 
State   of  Pennsylvania;   how    many    more    inhabitants 
were  there  in  New  York  than  in  Pennsylvania? 

Ans.  704888. 

45.  In  1880  the  number  of  male  persons  in  Louisiana 
was   468233,  and   the  number  of  females  471271 ;   how- 
many  more  females  than  males  were  there?     Ans.  3038. 

46.  In  1840  the  population  of  the  U.  S.  was  17069453, 
and  in  1880  it  was  50155783;  what  was  the  increase  in 
40  years?  Ans.  33086330. 


84  Intermediate  Arithmetic. 

89.  47.  Three  of  the  parts  of  1250  are  231,  365,  and 
189;  what  is  the  c  part? 

EXPLANATION. — Since  the  c  part  added  to          OPERATION. 

the  sum  of  the  other  parts   make   the  whole  231 

1250,   by  subtracting  the  sum   of  the  given  o^- 

parts   from   the   whole    we   get   the   c  part.  1 QQ 
Hence,  to  find  the  c  part  when  the  whole 

and  several  parts  are   known,   add  the  given  785      465  Ans. 
parts  and  subtract  their  sum  from  the  whole. 

48.  Two  of  the  parts  of  35  are   12   and   15 ;    what  is 
the  c  part? 

49.  Two  of  the  parts  of  $980  are  $435  and  $386;  what 
is  the  c  part?  Ans.  $159. 

50.  Three  of  the  parts  of  $1000  are  $555,  $222,  $111; 
what  is  the  c  part?  Ans.  $112. 

51.  The  sum  of  three  numbers  is  1160;  the  first  num- 
ber is  384,  the  second  571,  what  is  the  third?     Ans.  ? 

52.  A  boy  gathered  769  nuts,  of  which   he  gave  his 
sister  263  and  his  mother  378;  how  many  nuts  had  he 
left?  Ans.  128. 

53.  A  man  starts  on  a  journey  of  583  miles;  after  he 
travels  260  miles,  and   then  173  miles  more,  and   then 
95  miles  more ;  how  far  will  he  have  to  go  ? 

Ans.  55  miles. 

54.  A  man  owed  a  debt  of  $2000;  at  one  time  he  paid 
$520,  at  another  $763,  and  at  another  $391 ;  how  much 
does  he  still  owe?  Ans.  ? 

90.  Make  problems  of  the  following,  giving  the  answer 
to  each : 

55.  33296  c  — 22535  c  =  ? 

56.  50000  g  —  41715  g  =  ? 

57.  6760  d  — 3243  d  — 2189  ci  =  ? 

58    8713  b  +  1565  b  — 3767  b  — 4593  b  =  ? 


Subtraction.  85 

91.  PARALLEL  PROBLEMS. 

1.™  One  of  the  parts  of  25  is  12;  what  is  the  c  part? 

2.  A  and  B  have  together  $308,  of  which  A  owns 
8183.25;  how  many  dollars  has  B?  Am.  $124.75. 

3.™  Two  of  the  parts  of  33  are  11  and  9;  what  is  the 
C  part? 

4.  A  farmer  raised  3750  bushels  of  wheat,  corn  and 
barley,  of  which  1521  bushels  were  wheat,  and  1038 
bushels  corn ;  how  many  bushels  of  barley  did  he  raise  ? 

Ans.  1191. 

5.™  Jane  had  45  peaches,  of  which  she  gave  20  to 
Emma,  10  to  Rosa  and  7  to  Julia ;  how  many  did  Jane 
have  left? 

6.  A  man  having  $1768  on  deposit,  gave  a  check  for 
$175  to  A,  one  for  $238.25  to  B,  and  one  for  $369.50  to 
C ;  how  much  money  was  left  on  deposit  ?  Ans.  $985.25. 

7.™  James,  William  and  Henry  have  together  37 
oranges,  of  which  18  belong  to  James,  and  Henry  has 
9  oranges  less  than  James;  how  many  oranges  has 
William  ? 

8.  The  sum  of  three  numbers  is  6435 ;  the  first  is 
2816,  and  the  second  is  934  less  than  the  first ;  what  is 
the  third?  Ans.  1737. 

9.™  The  c  parts  of  a  number  are  12  and  13 ;  what  is 
the  number? 

10.  The  subtrahend  is  $425.15,  and  the  remainder 
$172.85:  what  is  the  minuend?  Ans.  8598. 

11.™  How  much  more  is  18  +  12  than  25 --6? 

12.  From  the  sum  of  783  and  248  subtract  the  differ- 
ence between  900  and  527.  Ans.  658. 

13. m  What  number  added  to  17  will  make  30? 

14.  One  of  the  parts  of  8305  is  6971;  what  is  the 
c  part?  Ans.  1334. 


86  Intermediate  Arithmetic. 

15. m  One  of  the  parts  of  20-fl5  is  10  -j-  5 ;  what  is 
the  C  part? 

16.  A  farmer  received  $2025  for  his  sugar  and  $1824 
for  his  cotton.  The  expense  of  raising  the  sugar  was 
$1113,  and  of  the  cotton  $749;  what  were  his  profits? 

Ans.  $1987. 

17. m  $13  and  $8  are  two  parts  of  $33;  what  is  the  c 
part? 

18.  A  man  paid  $5270  for  a  house  and  $1835  for  im- 
proving it.  If  he  sells  the  house  for  $7500,  what  will 
be  his  profits?  Ans.  $395. 

19.™  If  I  begin  at  40  and  count  backward,  what  will 
be  the  12th  number? 

20.     What  is  the  525th  number  below  937?    Ans.  412. 

21.™  50  contains  19,  11,  and  the  c  part;  what  is  the 
latter? 

22.  A  and  B  are  1529  feet  apart;  if  A  goes  toward 
B  375  feet,  and  B  goes  toward  A  682  feet,  how  far  apart 
will  they  then  be?  Ans.  472  feet. 

92.  QUESTIONS  FOR  REVIEW. 

What  is:  1.  Subtraction?  2.  The  minuend?  3.  The  subtra- 
hend? 4.  The  remainder?  5.  The  sign  of  Subtraction  ? 

What  is  denoted  by  the  sign  —  ? 

What  is  the  relation  of  Subtraction  to  Addition  ?  What  do  we 
find  by  Subtraction  ? 

What  is  the:  1.  Principle  of  Subtraction?  2.  Rule  for  Sub- 
traction? How  may  Subtraction  be  proved? 

When  we  know  the  whole  and  several  of  its  parts,  how  do  we 
find  the  c  part? 


MULTIPLICATION. 


INDUCTIVE  EXERCISES. 

93.   1.  How  many  ones  in  4  twos? 

*/ 

Aiis.   Two  taken  4  times,  or  2-j-2-}-2-f-2  =  8. 

2.  How  many  are  5  threes  ? 

Ans.  Three  taken  5  times,  or3-f3-f3-)-3-|-3  —  15. 

3.  How  many  are  4  sevens? 

Ans.  The  sum  of  four  sevens,  or  28. 

4.  How  many  are  : 

6  (5's)  ?    3  (9's)  ?    7  (4's)  ?     8  (5  dollars)  ?    5  (10  c.)  ? 
9  (10's)  ? 

5.  One  boy  has  two  hands;  how  many  hands  have  6 
boys?  Ans.  2  hands  taken  6  times,  or  12  hands. 

6.  A  horse  has  6  nails  in   each  of  his  4  shoes ;   how 
many  nails  in  all  ? 

Ans.  4  (6  nails),  or  6  nails   taken  as   many  times  as 
there  are  shoes. 

7.  What  will  5  hats  cost  at  $4  a  piece?     Ans.  5  ($4). 

8.  At  9c.  each,  what  will  three  melons  cost? 

9.  What  will  be  the  cost  of  6  pairs  of  boots  at  88  a 
pair  ?  Ans.  6  ($8)  =  ? 

10.  If  a  horse  travels  4  miles  per  hour,  how  far  does 
he  travel  in  7  hours  ? 

11.  There  are  5  trees  in  the  orchard  and  20  peaches  on 
each  tree ;  how  many  peaches  are  in  the   orchard  ?     20 
peaches  are  taken  how  many  times  ?    5  (20  peaches)  =  ? 

(87) 


88  Intermediate  Arithmetic. 

DEFINITIONS. 

94.  Multiplication  is  a  short  method  of  adding  equal 
parts,  or   the  process   of  taking   one   number  as  many 
times  as  there  are  units  in  another. 

95.  The  number  to  be  taken  is  called  the  multiplicand; 
the  number   that  shows  how   many   times  it  is  to  be 
taken,  the  multiplier,  and  the  result,  the  product. 

96.  The  multiplicand  and  multiplier  are  called  the  fac- 
tors of  the  product. 

97.  The  Sign  of  Multiplication  is  X  ,  which  is  read : 
times,  or  multiplied  by.     When  X  stands  between  two 
numbers,  it  indicates  that  one  of  them  is  to  be  multi- 
plied by  the  other. 

Thus,  6  X  5  ==  30  is  read  :  5  times  6  are  30,  or  6  multiplied  by 
5  =  30.  Here  6  is  the  multiplicand,  5  the  multiplier,  and  30  the 
product.  The  factors  of  30  are  5  and  6. 

Since  6X5  =  5X6,  either  may  be  read :  5  times  6  or  6  times  5. 
If  one  of  the  factors  is  a  concrete  number,  it  is  regarded  as  the 
multiplicand,  but  may  be  used  abstractly  as  the  multiplier.  It  is 
evident  that  5  (6)  also  indicates  multiplication.  See  Art.  69- 

98.  The  Complemental  Factors  of  a  number  are  those 
factors  whose  product  is  equal  to  that  number. 

Thus,  the  complemental  factors  of  6  are  2  and  3 ;  of  12,  3  and 
4,  or  2  and  6 ;  of  36,  4  and  9,  or  2,  3,  and  6,  etc.,  etc. 


99.  Copy  and  read: 

1.  4X3  =  12. 

2.  5  X  8  =  40. 

3.  9  X  4  =  ? 

4.  2X9=? 


5.6x2  yards  =  12  yards. 

6.  7  X  3  men    =21  men. 

7.  8  X  7  boys  =56  boys. 

8.  3X  8weeks  =  ? 


Point  out  the  multiplicand,  multiplier,  and  product  in 
each. 


Multiplication.  89 

100.  Express  by  signs: 


1.  5  times  7  are  35. 

2.  8  times  9  are  72. 

3.  7  times  $4  are  828. 


4.  4  taken  3  times  equals  12. 

5.  9  times  5  are  how  many  ? 

6.  6  times  4  pens  are  24  pens. 


101.  PRINCIPLES. 

1°.  The  multiplier  is  an  abstract  number. 

2°.  The  multiplicand  and  product  are  like  numbers. 

3°.  Multiplication  may  be  effected  by  Addition. 

Let  the  pupil  point  out  or  verify  each  of  these  principles  in 
examples  1,  2,  5,  6,  and  7  of  Art.  99- 

Thus,  in  ex.  5, 

1°.  6,  the  multiplier,  is  an  abstract  number. 

2°.  2  yards,  the  multiplicand,  and  12  yards,  the  product,  are  like 
numbers. 

3°.  6X2  yards  ==2  yards  +  2  yards  +  2  yards  +  2  yards  +  2 
yards  -f-  2  yards  ==12  yards. 

102.  By   Multiplication   we  find  a  number  when  we 
know  its  complemental  factors. 

Thus,  if  5  and  7  are  the  c  factors  of  a  number,  the  number 
is  5  X  7  =  35 ;  if  6  and  9  are  the  c  factors,  the  number  is  6  X 
9  =  54. 

SUGGESTIONS  TO  TEACHERS.— In  every  example  and  problem  in 
Multiplication  the  pupil  should  be  required  to  point  out : 

1°.  The  whole  and  the  c  parts. 

Thus,  in  4  X  $5  ==  ?,  the  parts  are  $5,  $-5,  $5,  $5,  and  the  whole 
is  the  answer  required,  viz :  $20. 

2°.  The  uftole  and  the  c  factors. 

Thus,  in  6X5=?,  the  c  factors  are  6  and  5,  and  the  whole  is 
the  answer  required,  30. 

103.  The  following  table  contains  the  products  of  each 
two  numbers  from  0  to  12.      It  should  be  thoroughly 
committed  to  memory. 


90 


Intermedia te  Arithmetic. 


MULTIPLICATION  TABLE. 


1 

2 

3 

4 

ix  0=    o 

2X    0=     0 

3X    0=     0 

4X    0=     0 

IX   1  =     1 

2X    1=     2 

3X    1=     3 

4X    1=     4 

IX    2  =     2 

2X    2=     4 

3X    2=     6 

4X    2=     8 

IX    3=     3 

2X    3=     6 

3X    3=     9 

4X    3=    12 

IX    4  =     4 

2X    4=     8 

3X    4=    12 

4X    4=    16 

IX    5-     5 

2X    5=    10 

3X    5=    15 

4X    5=    20 

IX    6=     6 

2X    6=    12 

3X    6=    18 

4X    6=    24 

IX    7  =      7 

2X    7=    14 

3X    7=   21 

4X    7=    28 

IX    8=     8 

2X    8=    16 

3X    8=   24 

4X    8=    32 

1X9=     9 

2X    9=    18 

3X    9=   27 

4X    9=    36 

1X10=    10 

2X10=    20 

3X10=    30 

4  X  10  =    40 

ixn=  11 

2X11=    22 

3X11=    33 

4XH=    44 

1X12=    12 

2X12=    24 

3X12=   36 

4  X  12  =    48 

5 

6 

7 

8 

5X    0=     0 

6X    0=     0 

7X   0=     0 

sx  o=    o 

5X    1  =     5 

6X    1=     6 

7X    1=     7 

8X    1  =     8 

5X    2=    10 

6X    2=    12 

7X    2=    14 

8X    2=    16 

5X    3=    15 

6X    3=    18 

7X    3=   21 

8X    3=    24 

5X    4  =    20 

6  X    4  =    24 

7X    4=   28 

8X    4=    32 

5X    5=    25 

6X    5=    30 

?X    5=    35 

8X    5=    40 

5X    6=   30 

6X    0=    36 

7X    6=    42 

8X    6=    48 

5X    7  =   35 

6X    7=   42 

7X    7=    49 

8X    7=    56 

5X    8=   40 

6X    8=    48 

7X    8=    56 

8X    8=   64 

5X    9=   45 

6X    9=    54 

7X    9=   63 

8X    9=    72 

5X10=   50 

6X10=    60 

7X10=   70 

8X10=   80 

5XH=   55 

6XH=    66 

7XH=    77 

8XH=    88 

5X32=    60 

6X12=    72 

7X12=    84 

8  X  12  =    96 

9 

10 

11 

12 

9X    0=     0 

10  X    0=     0 

nx  o=    o 

12  X    0=     0 

9X    1=     9 

10  X    1=    10 

11  X   1=   11 

12  X    1=    12 

9X    2=    18 

10  X    2  =    20 

11  X    2=    22 

12  X    2=    24 

9X    3=   27 

10  X    3=   30 

11  X    3=    33 

12  X    3=    36 

9X    4=    36 

10  X    4=    40 

11  X    4  =    44 

12  X    4=    48 

9X    5=   45 

10  X    5=   50 

11  X    5=   55 

12  X    5=    60 

9X    6=   54 

10  X    6=   60 

11  X    6=    66 

12  X    6=    72 

9X    7=    63 

10  X    7=   70 

11  X    7=    77 

12  X    7=    84 

9X    8=    72 

10  X    8=   80 

11  X    8=   88 

12  X    8=    96 

9X    9=    81 

10  X    9=   90 

11  X    9=   99 

12X9  =  108 

9X10=    90 

10  X  10  =  100 

11x10  =  110 

12X10=120 

9X11=    99 

10x11  =  110 

11  X  11  =  121 

12X11  =  132 

9  X  12  =  108 

10X12  —  120 

11  X  12  =  ^132 

12  X  12  =  144 

Multiplication. 


91 


DRILL  EXERCISES. 

104.  With    the   Combination    Boards,    Art.    61,    the 
teacher  may  provide  abundant  drill  exercises  in  Multi- 
plication.    In    the    absence    of  these    the    following    is 
recommended : 

Draw  a  circle  on  the  board,  and  within 
it  write  the  figures  from  0  to  9  inclusive, 
as  in  the  diagram.  At  the  center  write 
6,  and  let  the  pupils  name  the  product  of 
6  by  each  of  the  other  figures  taken  in 
order  around  the  circle.  Then  erase  6, 
and  in  its  place  write  one  of  the  other 
figures  to  be  used  as  a  multiplier ;  and 
so  continue  until  all  the  combinations 
shall  have  been  reached. 

This  exercise  should  be  used  in  class  drill  until  every  student 
can  name  all  the  products  instantly.  This  circle  may  be  used 
with  equal  facility  in  drilling  pupils  in  Addition. 

105.  The   following  examples   should   be    solved   by 
Addition  and  Multiplication   until  the  relation  of  the 
operations  is  clearly  apprehended. 

1.  Multiply  9  by  7;  8  yards  by  5. 

2.  What  is  the  product  of:   8  and  6?    5  and  7? 

3.  7X7  =  ?    3x  8  =  ?    6X9  =  ? 

4.  What  is  the  number  whose  c  factors  are  9  and  8? 

5.  What  is  the  value  of  6  if  its  unit  is  $7  ?     6x7  =  ? 

6.  What  will  8  hats  cost  at  $5  apiece? 

MENTAL  EXERCISES. 

106.  What  is  the  product  of: 

4  and  2?  5  and  3?  3  and  4?  6  and  2?  5  and  6? 
4  and  7?  5  and  4?  4  and  5?  3  and  6?  5  and  7? 
4  and  8?  6  and  7?  4  and  6?  7  and  2? 


92 


In  tern  ted  ia  te  A  rithmetic. 


2.  Multiply  3  by  3;  7  by  7  ;  6  by  6;  8  by  8;  5  by  5; 
9  by  9;  4  by  4;  2  by  2;  6  by  8  ;  2  by  5 ;  6  by  3 ;  7  by  9  ; 
9  by  3 ;  8  by  2. 

3.5X9  =  ?  2X12=?  7X8  —  ?  3x7  —  ?  6X 
4  =  ?  9X2  =  ?  4X11—?  6X9  =  ?  2x10  —  ?  4 
X9  =  ?  5XH=? 

4.  What  is  the  number  whose  c  factors  are  3  and  4? 

7  and  10?     9  and  8?     7  and  11?     3  and  8?     4  and  12? 

8  and  11?    5  and  12?    3  and  2?    8  and  12?    9  and  11? 
8  and  10? 

5.  What  pairs  of  digits,  when   multiplied,   will  give 
the  product :  12  ?    16?    18?    24?    36? 

6.  What  is  the  value  of  7  if  its  unit  is  9?     7  nines 
are  how  many  ? 

7.  What  is  the  value  of  4  if  its  unit  is  8?    4  eights 
are  how  many? 

8.  What  is  the  value  of  6  if  its  unit  is:  3c?    $5?    7 
hats  ?    4  days  ?    9  ?    2  feet  ?    6  ?     8  horses  ? 


5X4+10  =  ? 
4X2+  8  =  ? 
5x6--10  =  ? 
4X7-  5  =  ? 
5  X  4  +  12  =  =  ? 
5x0+  3  =  ? 


3x6+13 

5  X  7  -  - 12 
4X8+11 

6  X  7  -  - 10 
4X6+  9 
7X0+  0 


7X2-5 
3X3  +  4 
6X6--3 
7X7  +  2 
9X9-1 
8X2  +  0 


NOTE.— Perform  the  multiplication  first.    Thus,  5  X  2  +  3  =  10 
+  3  =  13. 

EXERCISES  IN  MAKING  PROBLEMS. 

107.   i.  Make  a  problem  of:  5  bX3  =  ? 

Ans.  5  books  taken  3  times  are  how  many? 
Or,  If   1   barrel  holds  5  bushels,  how   many  bushels 
will  3  barrels  hold  ? 


Multiplication.  93 

Make  a  problem  of: 

2.  2  c  X  8  =  ?         Ans.    ? 

3,  7  h  X4  =  ?         Ans.    ? 


4.  6clx5=  =? 

5.  9  tX6  =  ?          Ans. 


6.  Make  a  problem  of:  6eX4-f-5  c^=? 

Ans.  How  many  are  4  times  6  cups,  and  5  cups? 

Or,  6  cups  taken  4  times  and   5   cups  more  are  how 
many? 

Make   problems  of  the  following,  giving  ansAvers  to 
each : 

7i3gX8_L.3g=  ?  9.  6sX7  —  5s  =  ? 

8.  5ax7-flOa  =  ?  10.  9bx8-3b  =  ? 

MENTAL  EXERCISES. 

108.  l.  Of  what  number  are  7  and  5   the  c  factors? 

2.  What  is  the  cost  of  8  barrels  of  flour  at  $7  a  barrel  ? 

ANALYSIS.— Since  1  barrel  cost  $7,  8  barrels  will  cost  8  times  $7, 
or  $-56. 

3.  One  bushel  contains  4  pecks;  how  many  pecks  in 
7  bushels? 

4.  Seven   days    in    a    week,  how    many    days    in    8 
weeks  ? 

5.  What  is  the  cost  of  9  ploughs  at  $11  each? 
6    What  is  the  cost  of  12  hats  at  $5  each? 

7.  A  horse  walks  5   miles  an  hour,  how  far  does  he 
travel  in  7  hours? 

8.  Ten  cents  are  a  dime  and  10  dimes  are  a  dollar ; 
how  many  cents  are  in  a  dollar? 

9.  Three  feet  are  a  yard  and  12  inches  a  foot ;  how 
many  inches  in  a  yard? 

10.  Seven  days  in  a  week  and  4  weeks  in  a  month, 
how  many  days  in  a  month? 


94  Intermediate  Arithmetic. 

11.  Forty  rods  in  a  rood  and  4  roods  in  an  acre,  how 
many  rods  in  an  acre? 

•/ 

12.  Four  farthings  in  a  penny  and  12  pence  in  a  shil- 
ling; how  many  farthings  in  a  shilling? 

13.  What  will  be  the  cost  of  4  barrels  of  flour  at  $10 
a  barrel? 

14.  At  $9  each,  what  will  be  the  cost  of:  5  sheep?     7 
sheep?     10  sheep?     6  hogs?     4  guns? 

15.  How  many  are  5  X  10?  Ans.  5  tens,  or  50. 

16.  How  many  are  6X  10?  Ans.  6  tens,  or  60. 

NOTE  1. — A  number  is  multiplied  by  10  by  annexing  one  0 
to  it. 

17.  How  much  will  10  cigars  cost  at:  5  cents  apiece? 
12  cents  apiece?     15  cents  apiece?     25  cents  apiece? 

18.  How  much  will   16  Ibs.  sugar  cost  at  10  cents  a 
pound  ? 

19.  There  are  10  dimes  in  SI,  how   many  dimes  in  : 
$3?    $8?    $27?    $54?    $120?    $255? 

20.  There  are  10  square  chains  in  1  acre  ;  how  many 
square  chains  in :    7  acres  ?     13  acres  ?     42  acres  ?     145 
acres  ? 

21.  At  $10  apiece,  what  will  be  the  cost  of:    9  hogs? 
15  saddles?     21  guns?    32  clocks?     130  sheep? 

22.  How  many  are  6X  100?       Ans.  6  hunds.,  or  600. 

NOTE  2.  —  A  number  is  multiplied  by  100  by  annexing  two 
O's  to  it. 

23.  What  will  100  cigars  cost  at  4  cents  apiece? 

24.  There  arc  100  cents  in  $1 ;   how  many  cents  in  : 
$2?     $19? 

25.  There   are   100  years   in   1   century ;    how   many 
years  in:  5  centuries?      11  centuries?     37  centuries? 

26.  At  $100  apiece,  what  will  be  the  cost  of:  13  horses? 
26  carriages?    30  organs?     143  gold  watches? 


Multiplication. 


95 


27.  How  many  are  7  X  1000?     Ans.  7  thous.,  or  7000. 

28.  How  many  are  7  X  10000? 

Ans.  7  tcn-thous.  or  70000. 

109,  RULE.  -  -  To  multiply  by  10,  100,  1000,  annex  as 
many  O's  to  the  multiplicand  as  there  are  O's  in  the  mul- 
tiplier. 

29.  There  are  1000  mills  in  1  dollar ;  how  many  mills 
in  82?     $3?    $9?    $12?    $63? 

30.  There  are  1000  ounces   in  1  cubic  foot  of  water  ; 
how  many  ounces  in  3  cubic  feet  of  water?     8  cubic 
feet?     137  cubic  feet? 


WRITTEN  EXERCISES. 

110.  CASE  I. — When  the  multiplier  is  a  digit. 
1.  Multiply  435  by  7. 


EXPLANATION. — Write  the  multiplier  under  the 
multiplicand,  as  in  the  margin,  and  beginning 
with  the  units,  we  say : 

7X5  ones  =  35  ones  =  35, 
7X3  tens  =  21  tens  =  210, 
7X4  hunds.=  2S  hunds.  =  2800, 

which  added  together  make  3045. 


5  = 


In  practice,  the  work  is  abbreviated  thus :  7  X 
35,  write  the  5  and  carry  3  ;  7  X  3  =  21  and 
3  make  24,  write  the  4  and  carry  2  ;  7  X  -4  =  28 
and  2  make  30,  which  write. 


Multiply  : 

2.  325  by  4. 

3.  647  by  8. 


OPERATION. 

435 

7 

35 

210 

2800 

3045 

OPERATION. 

435 

7 


3045 


Ans.  1300. 
Ans.  5176. 


4.  605  by  7.       Ans.  4235. 


5.  487  by  3. 

6.  792  by  6. 

7.  986  by  5.       Ans.  4930. 


An*.  1461. 
Ans.  4752. 


8.  In  a  similar  manner  solve  the  examples  in  Art.  69. 


96  Intermediate  Arithmetic. 

9.  Multiplicand  is     643,  multiplier  is  7,  product  =  ? 

10.  Multiplicand  is  $725,  multiplier  is  4,  product  =  ? 

11.  Multiplicand  is  6289c.,  multiplier  is  6,  product  =  ? 

12.  Multiplicand  is  287  tops,  multiplier  is  9,  product  —  ? 

13.  If  one   bale   of  'cotton  weighs   456   pounds,  what 
will  6  bales  weigh  ?  Ans.  2736  pounds. 

14.  There  are  640  acres  in  a  square  mile;  how  many 
acres  in  9  sq.  miles?  Ans.  5760  acres. 

15.  There  are  8  rows  in  an  orchard  and  123  trees  in 
each  row  ;  how  many  trees  in  the  orchard  ?  Ans.   ? 

16.  A  man  travels  7  miles;  how  many  yards  does  he 
travel,  there  being  1760  yards  in  a  mile? 

Ans.  12320  yards. 

17.  Which  will  cost  the  more  ;  137  yards  at  6  cents  a 
yard,  or  117  pounds  at  7  cents  a  pound?  Ans.  ? 

111.  CASE  II.  —  When  the   multiplier  is  any  number. 

1.  Multiply  523  by  746.  OPERATION. 

746 
EXPLANATION.  —  Write  one  of  the  factors  (gen- 

erally the  less)  under  the  other,  as  in  the  margin,  ^o 

and,  beginning  at  the  right,  we  say  :  2238 

3  ones       X  746  =  2238  ones  =    2238,  14920 

2  tens        X  746  =  =  1492  tens  =  14920,  373000 

5  hunds.  X  746  =  3730  Jiunds.  373000,  390158 

which  added  together  make  390158. 

OPERATION. 

In  practice,  the  work   is  done  thus:    Multiply  746 

746  by  3,  2,  5,  in  succession  as  in  CASE  I,  writ-  523 

ing  the  products  and  their  sum  as  in  the  margin. 


The  products  resulting  from  multiplying          1492 

the   multiplicand   by   the   ones,   tens,   hun-  3730 

dredSj  etc.,  of  the  multiplier  are  called  par-  390158 
tial  products. 


Multiplication. 


97 


From  the  preceding  examples  and  explanations  we 
derive  the 

RULE. — I.    Write  the  multiplier  under  the  multiplicand. 

II.  Begin  at  the  right,  multiply  the  multiplicand  by  the 
ones,  tens,  hundreds,  etc.,  of  the  multiplier,  placing  the  right 
hand  figure  of  each  partial  product  under  the  figure  of  the 
multiplier  used  to  obtain  it,  and  add  the  partial  products. 

PROOF. — Multiply  the  multiplier  by  the  multiplicand,  and 
if  the  product  is  the  same  as  before,  the  work  is  probably 
correct. 

In  this  manner  multiply  and  prove : 

(2)  (3)  (4)  (5) 


96 
15 

Ans.  1440 


85 
25 


125 

81 


542 

27 


Ans.  2125       Ans.  10125 


(6) 

785 

72 


(7) 
1283 
144 

Ans.  184752 


Ans.  56520 
Multiply : 

9.  155  bushels  by  102. 

10.  275  days  by  203. 

11.  1652  hours  by  205. 

12.  15  X  14=?   Ans.  210.    14. 

13.  25  X  18=?        Ans.  ?   15. 

16.  Multiply  4350  by  2700. 


Ans.  14634 

(8) 
845 
108 

Ans.  91260 


Ans.  15810  bushels. 
Ans.  ? 

Ans.  338660  hours. 
125  X  23  =  ?  Ans.  2875. 
168  X  41=?  Ans. 


112.  RULE.  —  Omit  the  O's  on  the  right  of 
the  multiplicand  and  multiplier,  multiply  the 
remaining  figures  together,  and  annex  the  O's 
omitted  to  the  result.  ' 


OPERATION. 

435 

27 


N.  I.-7. 


98  Intermediate  Arithmetic. 

Multiply  : 

17.  125  by  20.    Aw.  2500.        19.  760  by  40.  Ans.  ? 


18.  348  bv  30.  Ans.  10440. 


20.  950  by  60.     Ans.  ? 


81.  708000  by  6500.  Ans.  4602000000. 

22.  45600  by  3400.  Ans.  ? 

What  is  the  number  whose  c  factors  are  : 

23.  648  and  100?  Ans.  ? 

24.  21200  and  70?  Ans.   ? 

25.  487100  and  27000?  Ans.  13151700000. 

26.  359260  and  3040?  Ans.  1092150400. 

27.  A  hogshead  holds  63  gallons;  how  many  gallons 
do  49  hogsheads  hold?  Ans.  3087  gallons. 

28.  If  a  vessel  sails  169  miles  a  day,  how  many  miles 
will  she  sail  in  576  days?  Ans.  97344  miles. 

29.  If  a  regiment  of  soldiers  contains  1128  men,  how 
many  men  are  there  in  an  army  of  106  regiments? 

Ans.  119568  men. 

30.  What  is  the  weight  of  45  bales  of  cotton,    each 
bale  weighing  463  pounds?  Ans.  20835  pounds. 

31.  What  is  the  weight  of  74  hogsheads  of  sugar,  each 
hogshead  weighing  1395  pounds?     Ans.  103230  pounds. 

32.  If  a  carriage  wheel  revolves  419  times  in  going  a 
mile,   how   many     times    will     it    revolve   in   going    2 
miles?    Smiles?     100  miles?     340  miles?  Ans.  ? 

83.  What  will  27  horses  cost  at  £140  apiece? 

Ans.  £3780. 

34.  If  1  buggy  cost  8143,  what  will   be    the  cost  of  3 
buggies  ?     10  buggies  ?.    50  buggies  ?     140  buggies  ? 

35.  There  are  365  days  in  a  year;  how  many  days  are 
there  in  32  years?  Ans.  11680  days. 

36.  An  orchard  contains  240  peach  trees.     If  there  are 
on   an   average   135   peaches   on   each   tree,   how   many 
peaches  are  there  in  the  orchard  ?  An*.  32400. 


Multiplication. 


99 


37.  Suppose  a  book  to  contain  470  pages,  45  lines  on 
each  page,  and  50  letters  in  each  line,  how  many  letters 
in  the  book?  Ans.  1057500. 


113.  There  are  : 

GO  pounds  in  1  bushel  of  wheat. 
50  pounds  in  1  bushel  of  corn. 
32  pounds  in  1  bushel  of  oats. 
100  pounds  in  1  keg  of  nails. 

How  many  pounds  in  : 

38.  384  bushels  of  wheat? 

39.  250  bushels  of  oats? 

40.  213  kegs  of  nails? 

41.  148  barrels  of  beef? 

42.  97  barrels  of  salt? 

43.  56  barrels  of  flour? 

44.  534  bushels  of  corn  ? 


196  pounds  in  1  barrel  of  flour. 
200  Ibs.  in  1  bbl.  of  pork  or  beef. 
280  pounds  in  1  barrel  of  salt. 
180  pounds  in  1  barrel  of  coal. 


Ans.  23040  pounds. 

Ans.  8000  pounds. 

Ans.  ? 

Ans.  29600  pounds. 
Ans.  ? 
Ans.  ? 
Am.  ? 
Ans.  155160  pounds. 


45.  862  barrels  of  coal? 

114.  Make  problems  of  the  following,  taking  the  terms 
in  order,  and  giving  the  answer  to  each  : 

46.  420  hX  16+180  h  =  ? 

47.  518  cx  61-  -2963  c=? 

48.  6080  d  X  360  —  12384  d  =  ? 


115.  PARALLEL  PROBLEMS. 

i.m  What  is  the  number  whose  C  factors  are  12  and  7? 
2.     What  is  the  product  of  452  and  25?       Ans.  11300. 
3.m  What  will  9  oranges  cost  at  8  cents  apiece  ? 
4.     What  will  136  tables  cost  at  818  per  table? 

Ans.  $2448. 
5.™  How  long  will  it  take  1  man  to  do  a  work  which 

•v 

8  men  can  do  in  12  days? 


100  Intermediate  Arithmetic. 

6.  If  54  men  can  build  a  wall  in  36  days,  how  long 
will  it  take  1  man  to  build  it?  Ans.  1944  days. 

7.™  How  many  units  in  7?  If  each  of  them  is  5  cts., 
what  will  all  of  them  amount  to? 

8.  If  each  unit  of  36-5  is  24  hours,  how  many  hours 
are  there  in  all?  Ans.  8760  hours. 

9.™  What  is  the  number  whose  c  parts  are  $19  and 
12  times  $5? 

10.  If  a  boy  has  $93  when  he  is  18  years  old,  and 
saves  $125  each  year  until  he  is  30  years  old,  how  much 
money  will  he  then  have?  Ans.  $1593. 

ll.m  If  a  boy  gathers  16  nuts  each  day  and  eats  7  of 
them  each  night,  how  many  nuts  will  he  have  in  6 
days? 

12.  A  government  surveyor  receives  $150  a  month, 
and  expends  $72;  how  much  does  he  save  in  9  months? 

Ans.  $702. 

13. "i  A  father  has  5  sons;  to  2  of  them  he  gave  $8 
apiece,  and  to  each  of  the  others  $9;  how  much  did  he 
give  them  all  ? 

14.  23  boys  went  out  to  gather  chestnuts.  They 
separated  into  two  squads,  15  boys  going  in  the  first 
squad.  Each  boy  of  the  first  squad  gathered  324  chest- 
nuts, and  each  of  the  second  245  chestnuts  ;  how  many 
chestnuts  were  gathered  in  all  ?  Ans.  6820  chestnuts. 

15. "i  A  boy  had  5  boxes,  and  in  each  box  5  oranges, 
and  sold  each  orange  for  5  cents ;  how  much  did  he  re- 
ceive for  all? 

16.  A  load  of  12  bales  of  cotton,  each  bale  weighing 
450  pounds,  was  sold  at  9c.  a  pound  ;  what  was  the 
sum  received  for  the  load  ?  Ans.  48600  cents. 

I7.m  What  is  the  number  whose  c  parts  are  9  times 

and  $15  ? 

18.     A  drover  bought   48  sheep  at  $2.50  a  head,  and 


Multiplication.  101 

sold  them  for  $23.25   more  than  the  cost;  what  did  he 
receive  for  them?  Am.  $143.25. 

I9.m  What  is  the  number  whose  three  c  parts  are  2 
times  3,  4  times  5,  and  6  times  10? 

20.  A  merchant  bought  24  sets  of  crockery  of  45 
pieces  each,  29  sets  of  37  pieces  each,  and  54  sets  of  60 
pieces  each;  how  many  pieces  did  he  buy?  -4ns.  5393. 

2l.mlf  John  eats  7  biscuits  in  one  day,  how  many 
will  he  eat  in  a  month  of  30  days? 

22.  A  common  clock  strikes  156  times  every  day ; 
how  many  times  does  it  strike  in  a  year  of  365  days? 

Ans.  56940. 

23. m  How  much  more  is  12  times  5  than  28  and  17 
added  together? 

24.  What  is  the  difference  between  the  product  of  375 
and  120,  and  the  sum  of  28507  and  13629?  Ans.  2864. 

25. m  If  the  multiplicand  is  the  sum  of  8  and  7,  and 
the  multiplier  the  difference  between  13  and  4,  what  is 
is  the  product  ? 

26.     (381  +  264)  X  (583  —  196)  =  ?          ^  Ans.  249615. 

116.  QUESTIONS  FOR  REVIEW. 

What  is:  1.  Multiplication?  2.  The  multiplicand?  3.  The 
multiplier  ?  4.  The  product  ?  5.  The  Sign  of  Multiplication  ? 

What  is  denoted  by  the  sign  X  ? 

State  the  three  principles  of  Multiplication. 

What  are  the  c  factors  of  a  number?  What  do  we  find  by 
Multiplication  ? 

How  is  a  number  multiplied  by  10,  100,  etc.  ? 

What  is  the  rule  for  Multiplication  when  the  multiplier  is :  1. 
A  digit?  2.  Any  whole  number? 


DIVISION. 


INDUCTIVE  EXERCISES. 

117.  Are  3  and  3  complemental  parts  of  6?     Docs  6  , 
contain  its  parts  ?    How  many  3's  does  6  contain  ?    3  is 
contained  in  6  how  many  times  ?    How  many  times  can 
3  be  subtracted  from  6? 

How  many  times  is  4  contained  in  20? 

Aiis.  As  many  times  as  4  can  be  subtracted  from  20. 

20  —  4  =  16,  1  time;    16 --4  =  =  12,  2  times;    12  —  4  =  8,  3  times; 
8  —  4  =  4,  4  times  ;  4  —  4  =  0,  5  times. 

How  many  times  is  3  contained  in  18?  5  contained 
in  30?  7  contained  in  28?  9  contained  in  45?  11  con- 
tained in  6G?  12  contained  in  84? 

A  man  divided  $8  equally  among  his  sons,  giving 
each  $2;  how  many  sons  had  he?  $2  is  contained  in 
$8  how  many  times? 

A  horse  traveled  28  miles  at  the  rate  of  4  miles  per 
hour;  how  many  hours  did  he  travel?  28  contains  4 
how  many  times? 

James  has  20  marbles  and  wishes  to  put  them  in 
boxes  so  as  to  have  5  marbles  in  each  box ;  how  many 
boxes  does  he  need?  20  contains  5  how  many  times? 

How  many  times  is  5  contained  in  17  ? 

17  —  5  =•  =  12,  1  time  ;  12  —  5  =  7,  2  times  ;  7  --  5  =  2,  3  times. 

Ans.  3  times  and  2  over. 

How  many  times  is  4  contained  in  23  ?  6  in  31  ?  7 
in  46?  8  in  27?  9  in  33? 

(102) 


Division.  103 

DEFINITIONS 

118.  Division  is  the  process  of  finding  how  many  times 
one  number  is  contained  in  another. 

119.  The  number  to  be  divided  is  the  Dividend,  the 
number   by   which   it   is   divided   the   Divisor,  and  the 
result  the  Quotient. 

120.  When  a  part  of  the  dividend  is  left  after  the  di- 
vision is  performed,  it  is  the  Remainder,  and  must  al- 
ways be  less  than  the  divisor. 

121.  The  Sign  of  Division  is  -K     It  is  read :  divided  by, 
and  shows  that  the  number  before  it  is  to  be  divided  by 
the  number  after  it. 

Thus,  35  H-  7  -  =  5  is  read  :  35  divided  by  7  equals  5.  This  is  also 
read :  35  contains  7  5  times.  Here,  35  is  the  dividend,  7  the  divisor, 
and  5  the  quotient.  35-^-7  is  sometimes  written  7)35,  and  ^. 


122.  Copy  and  read: 

1.  12-5-3  =  4. 

2.  9)18  =  2. 


4.  24  men-^6  men=4. 

5.  56  boys-^-  7  =  8  boys. 
3.  J^  =  6.                          6.  3)27  weeks  =  9  weeks. 

Point  out  the  dividend,  divisor,  and  quotient  in  each. 
123.  Express  by  signs : 


1.  35  contains  7,  5  times. 

2.  72  contains  8,  9  times. 


3.  $28  contains  $4,  7  times. 

4.  12  divided  by  4  is  3. 


5.  45  contains  9  how  many  times? 

6.  24  pens  divided  by  6  equals  4  pens. 

124.  PRINCIPLES. 

1°.  When  the  divisor  and  dividend  are  like  numbers, 
the  quotient  is  an  abstract  number. 


104  Intermediate  Arithmetic. 

2°.  When  the  divisor  is  an  abstract  number,  the  quo- 
tient and  dividend  are  like  numbers. 

3°.  When  the  divisor  and  dividend  are  like  numbers, 
division  may  be  effected  by  subtraction. 

Let  the  pupil  point  out,  or  verify,  these  principles  in  each 
example  of  Art.  122. 

RELATION  OF  DIVISION  TO  MULTIPLICATION. 

125.  4  times  3  are  how  many?     3  is  contained  in  12 
how  many  times?   4  is  contained  in  12  how  many  times? 

4  times  what  are  12  ?  Ans.  3,  because  12  -5-  4  =  3. 
What  times  3  are  12  ?          Ans.  4,  because  12  H-  3  =  4. 

QUESTIONS  AND  ANSWERS. 

5  x  ?  =  20?  Ans.  4,  because  20-^5  =  4. 
?  X  6  =  42  ?                          Ans.  7,  because  42  -*-  6  =  7. 

In  a  similar  manner  answer  the  following,  perform- 
ing the  division,  if  necessary,  by  subtraction : 

7  X  ?  =  21.         ?  X  6  =  42.         9  men  X  ?  =  36  men. 
5  X  ?  =  30.        8  X  ?  =  56.         ?  X  4  cents  =  32  cents. 

PRINCIPLES. 

1°.  Division  is  the  reverse  of  multiplication. 

2°.  By  division  we  find  one  of  the  complemental  fac- 
tors of  a  number,  when  the  number  and  the  other  fac- 
tor are  given. 

126.  Since   Division   is  the  reverse  of  multiplication, 
by  reversing  the  table  of  the  latter,  we   have  the  fol- 
lowing table,  which  may  be  read  thus  :   1  in  none,  no 
times  ;  1  in  1,  one  time  ;  1  in  2,  two  times,  ete. 


Division. 


105 


DIVISION  TABLE 


1 

2 

3 

4 

1  in   0-  0 

2  in   0  =  0 

3  in   0=0 

4  in   0=0 

1*  in   1=1 

2  in   2  -   1 

3  in   3=1 

4  in   4  =   1 

1  in   2  =  2 

2  in   4  =   2 

3  in   6=   2 

4  in   8=2 

1  in   3=  3 

2  in   6=3 

3  in   9  =   3 

4  in  12=  3 

1  in   4=4 

2  in   8=4 

3  in  12  =   4 

4  in  16=   4 

1  in   5=5 

2  in  10=  5 

3  in  15  =  5 

4  in  20=  5 

1  in   6=6 

2  in  12=  6 

3  in  18  =  6 

4  in  24=  6 

1  in   7=7 

2  in  14=  7 

3  in  21  =  7 

4  in  28=  7 

1  in   8=8 

2  in  16=  8 

3  in  24=  8 

4  in  32  =   8 

1  in   9=9 

2  in  18=  9 

3  in  27  =  9 

4  in  36  =  9 

1  in  10  ==10 

2  in  20  =  10 

3  in  30  =  10 

4  in  40  =  =  10 

1  in  11  =  11 

2  in  22  =  11 

3  in  33  =  11 

4  in  44  =  11 

1  in  12  =  12 

2  in  24  =  12 

3  in  36  =  12 

4  in  48  =  12 

5 

6 

7 

8 

5  in   0  -   0 

6  in   0=0 

7  in   0  =  0 

8  in   0=0 

5  in   5=1 

6  in   6=1 

7  in   7=1 

8  in   8=1 

5  in  10  =  2 

6  in  12=  2 

7  in  14=   2 

8  in  16=  2 

5  in  15=  3 

6  in  18=  3 

7  in  21  =  3 

8  in  24=  3 

5  in  20=  4 

6  in  24  =  4 

7  in  28=  4 

8  in  32=  4 

5  in  25  =  5 

6  in  30  =  5 

7  in  35=  5 

8  in  40  =  5 

5  in  30=  6 

6  in  36=  6 

7  in  42=   6 

8  in  48  =  6 

5  in  •  35  =  7 

6  in  42=  7 

7  in  49=  7 

8  in  56=  7 

5  in  40  =  8 

6  in  48=  8 

7  in  56=  8 

8  in  64=  8 

5  in  45  =  9 

6  in  54  =  9 

7  in  63  =   9 

8  in  72=  9 

5-  in  50  =  10 

6  in  60  =  =  10 

7  in  70  =  ;  10 

8  in  80  =  10 

5  in  55  =  11 

6  in  66  =  11 

7  in  77  =  --  11 

.  8  in  88  =  11 

5  in  60  =  12 

6  in  72  =  =  12 

7  in  84  =  r  12 

8  in  96  =  12 

9 

10 

11 

12 

9  in   0=0 

10  in   0=0 

11  in   0  =   0 

12  in   0=0 

9  in   9=1 

10  in  10=  1 

11  in  11=  1 

12  in  12  =  1 

9  in  18=  2 

10  in  20  =  2 

11  in  22=  2 

12  in  24  =  2 

9  in  27  =  3 

10  in  30  =  3 

11  in  33  =  3 

12  in  36  =  3 

9  in  36  =  4 

10  in  40  =  4 

11  in  44=  4 

12  in  48  =   4 

9  in  45  =  5 

10  in  50  =  5 

11  in  55=  5 

12  in  60=  5 

9  in  54=  6 

10  in  60  =  6 

11  in  66=  6 

12  in  72  =  6 

9  in  63  =  7 

10  in  70  =   7 

11  in  77=  7 

12  in  84  =  7 

9  in  72=  8 

10  in  80=  8 

11  in  88=  8 

12  in  96  =  8 

9  in  81  =  9 

10  in  90  =  9 

11  in  99=  9 

12  in  108=  9 

9  in  90  =  10 

10  in  100  =  10 

11  in  110  =  10 

12  in  120  =  10 

9  in  99  =  11 

10  in  110  =  11 

11  in  121  =  11 

12  in  132  =  11 

9  in  108  =  12 

10  in  120  ==12 

11  in  132  =  =  12 

12  in  144  =  =  12 

106  Intermediate  Arithmetic. 


* 

DRILL  EXERCISES. 


127.  These  exercises  should  be  studied  by  the  pupils. 
They  should  also  be  written  on  the  board,  and  used  in 
class  drill  daily,  until  every  pupil  can  call  all  the  quo- 
tients instantly. 


2)4_ 

10 

18 

_6 

12 

_8 

14 

20 

16 

2 

3)3 

27 

9 

21 

15 

12 

18 

_6 

24 

30 

4)12 

28 

20 

16 

24 

_8 

32 

40 

_4 

36 

5)45 

15 

35 

25 

20 

30 

10 

40 

50 

_5 

6)60 

48 

12 

36 

24 

30 

42 

18 

54 

_6 

7)28 

42 

14 

56 

70 

35 

_7 

63 

21 

49 

8)24 

56 

40 

32 

48 

16 

64 

80 

_8 

72 

9)90 

72 

18 

54 

36 

45 

63 

27 

63 

_9 

MENTAL  EXERCISES. 

128.  These  exercises  should  be  performed  by  Subtrac- 
tion and  Division  until  the  relation  of  the  two  opera- 
tions is  clearly  apprehended. 

CASE  I. — When  there  is  no  remainder. 

1.  How   many  2's  in  4?     3's  in   3?     4's  in   12?    5's 
in  45?     6's  in  60?     7's  in  28?     8's   in  24?     9's  in  90? 

2.  Divide  10  by  2;  27  by  3;   28  by  4;    15  by  5;   48 
by  6 ;  42  by  7 ;  56  by  8 ;  72  by  9. 

3.2)2=?     2)18  —  ?     3)9-=?     4)20  =  ?    5)35=? 

4.  f  =  ?    ^-  =  ?    ^  =  ?    ¥  =  ?   ¥  =  ?   ¥  = ? 

6.  2x?  =  12.  3X?  =  15.  4X?=36.  5  X  ?  —  20. 
6  X  ?  =  24.  ?  X  7  =  =  70.  ?  X  8  =  :  48.  ?  X  9  =•-  36.  ?  X 
5  =  15.  ?  X  7  =  49. 


Division.  107 

6.  How  many  times  7  feet  make  42  feet? 

7.  How  many  times  3  days  make  15  days? 

8.  How  many  times  are  6  cents  contained  in  54  cents? 

9.  How   many  times   are   8   bushels   contained    in  56 
bushels  ? 

10.  I  bought  30   cents  worth  of  oranges   and   paid  5 
cents  for  each  orange  ;  how  many  did  I  l)uy  ? 

11.  If  6  yards  of  cloth  make  1  suit,  how  many  suits 
can  be  made  of  60  yards? 

12.  How  many  hours  will  it  take  a  man  to  walk  21 
miles,  if  he  walks  3  miles  per  hour? 

13.  There  are  40  boys  in  school,  and  8  boys  in  each 
class ;  how  many  classes  are  there  ? 

14.  The  dividend   is  54,   the   divisor   6;   what  is  the 
quotient? 

15.  The   dividend   is   63,  the   divisor   9;    what  is  the 
quotient  ? 

16.  8  is  a  factor  of  40;  what  is  the  c  factor? 

17.  7  is  a  factor  of  42;  what  is  the  c  factor? 
129.  CASE  II. — When  there  is  a  remainder. 

18.  How  many  times  is  8  contained  in  51? 

OPERATION.— The  next  number  below  51  that  8  will  divide 
evenly  is  48,  which  contains  8  6  times ;  and  since  51  is  3  more 
than  48,  we  say :  8  is  contained  in  51  6  times  and  3  over. 

How  many  times  is : 

19.  2  contained  in    5?  3  in  5    ?  4  in  15  ?  5  in  49? 

20.  6  contained  in  65  ?  7  in  34?  8  in  31?  9  in  98? 

21.  2  contained  in  11?  3  in  28?  4  in  30?  5  in  17? 

22.  6  contained  in  50?  7  in  45?  8  in  60?  9  in  80? 

23.  2  contained  in    9?  3  in  14?  4  in  11?  5  in  32? 

24.  6  contained  in  35  ?  7  in  39  ?  8  in  21  ?  9  in  47  ? 

25.  2  contained  in  21?  3  in    7?  4  in  43?  5  in  43? 

26.  6  contained  in  23?  7  in  67?  8  in  86?  9  in  33? 


108  Intermediate  Arithmetic. 

EXERCISES  IN  MAKING  PROBLEMS. 

130.   l.  Make  a  problem  of:  10  d-=-5==? 

Ans.  10  dollars  divided  by  5  equals  how  many? 
Or,  If  5  apples  cost  10  dimes,  how  much  will  1  apple 
cost? 

Make  a  problem  of: 


2.  20  c-5-4  =  ? 

3.  42  m  -=-  6  =  ? 


4.  63  a  ^7:  :? 

5.  54  b  -*-9  =  ? 


6.  Make  a  problem  of:  10  cX4--8c=:? 
Ans.  10  cups  taken  4  times  contain  8  cups  how  many 
times  ? 

Make  a  problem  of: 

7.  9  b  X  4-f-6  b  — ?  Ans.  ? 

8.  8  g  X  6-^-12  g=?  Ans.  ? 

9.  Make  a  problem  of:  7  b  X  4  +  2  b-^-5  b  =? 
Ans.  7  books  taken  4   times  and   2  books   more  con- 
tain 5  books  how  many  times? 

Make  problems  of  the  following,  taking  the  terms  in 
order : 

10.  9c    X2  +  2c    -j-5  c    =?  Ans.  4. 

11.  4mX5--6m-f-7m  =  ?  Ans.  2. 

12.  9hX7  +  3h-5-6h=?  Ans.  ? 

13.  10  d    X  4  — 5  d  -f-7  d  =?  Ana.  ? 

EQUAL  PARTS. 

131.  When  a  number  is  divided : 

Into  two  equal  parts,  one  of  the  parts  is  1  half  of  the 
number ; 

Into  three  equal  parts,  one  of  the  parts  is  1  third  of 
the  number; 


Division.  109 

Into  four  equal  parts,  one  of  the  parts  is  1  fourth  of 
the  number; 

Into  five  equal  parts,  one  of  the  parts  is  1  fifth  of 
the  number,  and  so  on. 

The  parts  1  half,  1  third,  1  fourth,  1  fifth,  etc.,  are 
written  :  £,  |,  J,  i,  etc. 

To  find  1  half  of  a  number  :  Divide  the  number  by  2. 

To  find  1  third  of  a  number :  Divide  the  number  by  3. 

To  find  1  fourth  of  a  number  :  Divide  the  number  by  4. 

To  find  1  fifth  of  a  number :  Divide  the  number  by  5. 

MENTAL  EXERCISES. 

132.  i.  How  much  is:  1  half  of  16?  1  third  of  24? 
1  fourth  of  36?  1  fifth  of  50?  1  sixth  of  54?  1  seventh 
of  21?  1  eighth  of  72?  1  ninth  of  27?  1  half  of  14? 
1  third  of  18  ?  1  fourth  of  32  ?  1  fifth  of  30  ?  1  sixth 
of  42  ?  1  seventh  of  70  ? 

2.  How  much  is  :  £  of  12  ?    J  of  15  ?     i  of  24  ?     J  of 
20?    £of36?    |  of  49?    |  of  48?    J  of  54? 

3.  6  feet  are   in   2   yards;   how   many   feet   are   in   1 
yard?  Ans.  j-  of  6  feet,  or  3  feet. 

4.  20  inches  in  4  hands;  how  many  inches  in  1  hand? 

5.  35  days  in  5  weeks;  how  many  days  in  1  week? 

6.  30  yards  in  6  suits;  how  many  yards  in  1  suit? 

7.  42  chairs  in  7  sets;  how  many  chairs  in  1  set? 

8.  If  8  hats  cost  $24,  what  is  the  cost  of  1  hat? 

9.  If  7  guns  cost  $63,  what  is  the  cost  of  1  gun  ? 

10.  If  6  peaches  cost  12   cents,  what  is  the  cost  of  1 
peach  ? 

11.  If  9  pounds  of  sugar  cost  90  cents,  what   is  the 
cost  of  1  pound? 

12.  If  10  gallons  of  whisky  cost  $30,  what  is  the  cost 
of  1  gallon  ? 


110  Intermediate  Arithmetic. 

13.  If  a  horse  walks  42  miles  in  7  hours,  how  many 
miles  does  he  travel  in  1  hour? 

14.  If  8  men  earn   $32   in  a   day,  how  many  dollars 
does  1  man  earn  ? 

15.  If  12  pairs  of  shoes  cost  $36,  what  will  1  pair  of 
shoes  cost? 

16.  Five  boys  gathered   40  pints  of  chestnuts,  which 
they  shared  equally ;  how   many   pints  had  each  boy  ? 

17.  How  far   must   a   man   travel   per   hour  to  go  24 
miles  in  6  hours? 

18.  If  10  acres  produce  100  bushels,  how  many  bushels 
is  that  per  acre? 

19.  How  much  is  one-sixth  of  30  feet? 

20.  One-fifth  of  a  pole   35  feet  long  was   broken   off; 
how  many  feet  were  broken  off? 

21.  Is  5  one-fifth  or  one-fourth  of  20? 

22.  If  8  yards  of  cloth  cost  56  cents,  what  is  the  cost 
of  1  yard? 

23.  If  9  hats  cost  $36,  what  is  the  cost  of  1  hat? 

24.  If  12    horses   consume   72   bushels  of  corn,   how 
much  will  1  horse  consume  in  the  same  time? 

25.  A  fox  is  54  feet  ahead  of  a  hound;  if  the  hound 
gains  on   him   6   feet   in   every  minute,  in   how   many 
minutes  will  he  overtake  the  fox? 

26.  Two   boats    on    the    Mississippi   R.   are  50   miles 
apart.     The  hindmost  boat  gains  on   the  other  5  miles 
an    hour ;    in    how    many    hours   will    it    overtake    the 
other? 

27.  If  1  pipe  discharges  a  cistern  of  water  in  84  hours, 
how  long  will  it  take  7   pipes  of  the  same   size   to  dis- 
charge the  cistern? 

28.  4  times  9  are  how  many  times  6?     12?     3? 

29.  6  times  8  are  how  many  times  12?     4? 

30.  How  many  times  is   10  contained   in  5  times  6? 


Division.  Ill 

31.  How  many  times  is   6   contained  in  5  times  12? 

32.  How  many  times  is  4  contained  in  19 -f  5? 

33.  How  many  times  is  7  contained  in  40- -5? 

34.  How  many  times  is  3  contained  in  21  ?    $4  con- 
tained  in   $28?    5    bushels   contained   in   45    bushels? 
6  tens  contained  in  42  tens?     7  a  contained  in  56  a? 
9    u    contained    in    .108   u?     8    fifths  contained    in    64 
fifths  ? 

35.  How  much  is  1  fourth  of  16?     Of  $20?     Of  20  u? 

36.  How  much  is  £  of  15c.  ?    £  of  18  cows?    |  of  28 
b?    |   of  9  men?    |   of  9   tens?    £   of  9   u?    i   of  9 
sevenths  f 

WRITTEN  EXERCISES. 

133.  CASE  I. — When  each  figure  of  the  dividend  is 
divisible  by  the  divisor. 

1.  Divide  369  by  3. 

OPERATION. 

EXPLANATION.— 369  —  3  hunds.,  6  tens,  \  ones, 
which  divided  by  3  gives  1  hund.,  2  tew,  3  ones, 
or  123.  123 

Hence,  the 

RULE. — Place  the  divisor  on  the  left  of  the  dividend,  di- 
vide each  figure  of  the  latter  by  the  former,  and  write  the 
quotient  under  the  figure  divided. 

Divide : 


2.  264  by  2.         Ana.  132. 

3.  84  by  4.         Ans.    21. 

4.  906  by  3.         Ans.  302. 


5.  3063  by  3.     Ans.  1021. 

6.  4804  by  4.  Ans.  ? 

7.  2846  by  2.  Ans.  ? 


8.  What  will  1  book  cost  if  3  books  cost  63  cents? 

Ans.  21  cents. 

9.  What  will  one  lot  cost  if  2  lots  cost  $248? 

Ans.  $124. 


112  Intermediate  Arithmetic. 

10.  How  far  does  a  railroad  train  go  in  1  hour  if  it 
travels  96  miles  in  3  hours?  Ans.  32  miles. 

11.  A  father  divided  8084   chestnuts  equally   among 
4  boys ;  how  many  did  each  boy  get  ?  Ans.  ? 

134.  CASE  II. — When  each  figure  of  the  dividend  can 
not  be  divided  evenly  by  the  divisor. 

l.  Divide  11352  by  3. 

Now,  we  may  arrange  the  dividend  so  that  the  tens, 
hunds.,  etc.,  shall  each  be  divisible  by  the  divisor,  and 
then  divide  as  in  CASE  I. 

OPEKATION. 

11352  —  11  thous.     3  hunds.     5  tens,  2  ones. 

—    9  thous.  23  hunds.     5  tens,  2  ones. 

=-   9  thous.  21  hunds.  25  tens,  2  ones. 

=   9  thous.  21  hunds.  24  tens,  12  ones. 

3)9  thous.  21  hunds.  24  tens,  12  ones. 
3  thous.     7  hunds.     8  tens,     4  ones  =  3784. 

EXPLANATION. — The  first  number  below  11  thous.  that  is  divis- 
ible by  3  is  9  thous.,  which  we  write  in  the  place  of  11  thous.  and 
prefix  the  2  over  to  3,  making  23  hunds. 

Again,  the  first  number  below  23  hunds.  that  is  divisible  by  3 
is  21  hunds.,  which  we  write  in  the  place  of  23  hunds.,  and  pre- 
fix the  2  over  to  5,  making  25  tens.  Again,  the  first  number  be- 
low 25  tens  that  is  divisible  by  3  is  24  tens,  which  we  write  in  the 
place  of  25  tens,  and  prefix  the  1  over  over  to  2,  making  12  ones. 
Now  dividing  by  3,  as  in  CASE  I,  we  have  3  thous.  7  hunds.  8 
tens.  4  ones,  or  3784. 

SUGGESTIONS  TO  THE  TEACHER. — This  method  of  performing  Di- 
vision is  a  splendid  exercise  for  beginners.  It  not  only  prepares 
them  for  the  mechanical  operations  of  Division,  but  leads  them 
into  a  clear  conception  of  the  principles  on  which  the  operations 
depend.  The  parenthesis  may  be  used  to  separate  the  tens,  hun- 
dreds, etc.,  if  preferred,  as  in  the  following  example : 


Division. 


113 


2.  Divide  13572  by  4. 

OPERATION. 

1st  arrangement,             (13)  (  5)   (  7)   (  2) 

2d                                       (12)  (15)   (  7)   (  2) 

3d            "                        (12)  (12)   (37)   (  2) 

4th          "                         (12)  (12)   (36)   (12) 

Dividing  through  by  4,     3  3       9        3,    or  3393,  Ans. 
Divide : 


3.  108  by  3. 

4.  172  by  4. 

5.  280  by  5. 

6.  342  by  6. 


Am.  36. 
Ans.  43. 
Ans.  56. 
Ans.  57. 


7.  1401  by  3. 

8.  4325  by  5. 

9.  7230  by  10. 
10.  34120  by  8. 


Ans.  467. 
Ans.  865. 
Ans.  723. 
Am.  ? 


SHORT  DIVISION. 


43065  Rem.   1. 


135.  When  the  divisor  is  12  or  less,  we  use  short  di- 
vision. That  is,  we  perform  the  operations  mentally, 
and  write  the  results  only. 

l.  Divide  387586  by  9. 

OPERATION. 

EXPLANATION.  -  -  Write  the  divisor  and 
dividend  as  in  the  margin.  We  now  say  9 
in  38,  4  times  and  2  over  ;  write  4  below  and 
prefix  2  to  7,  making  27.  Again,  9  in  27,  3 

times  and  0  over  ;  write  3  below,  and  nothing  to  carry  to  the  5. 
Hence,  9  in  5,  0  times  and  5  over  ;  write  0  below  and  prefix  5  to  8, 
making  58.  Again,  9  in  58,  6  times  and  4  over  ;  write  6  below  and 
prefix  4  to  6,  making  4G.  Again,  9  in  46,  5  times  and  1  over  ;  write 
5  below  and.  1  Rem.  to  the  right.  The  pupil  should  be  trained 
to  call  results  only.  Thus,  4,  3,  0,  6,  5,  and  Rem.  1. 

From  the  preceding  articles  we  derive  the 

RULE  —  I.  Write  the  divisor  at  the  left  of  the  dividend 
with  a  line  between. 

N.  I.—  8. 


114 


Intermediate  Arithmetic. 


II.  Find  how  many  times  the  divisor  is  contained  in  the 
first  left-hand  figure  or  figures  of  the  dividend,  and  write  the 
quotient  underneath,  and  so  proceed  with  each  figure. 

III.  If  there  is  a  remainder,  prefix  it  to  the  next  figure 
of  the  dividend,  and  divide  as  before. 

IV.  When   the  divisor  is  not  contained  in  any  partial 
dividend,  write  a  cipher  in  the  quotient,  and  prefix  this  num- 
ber to  the  next  figure  of  the  dividend,  and  divide  as  before. 

PROOF.  —  Multiply  the  quotient  by  the  divisor,  and  to  the 
product  add  the  remainder,  if  any  ;  the  result  should  be  equal 
to  the  dividend. 


EXERCISES 


Divide  : 


2.  3754  by  2.        Ans.  1877. 

3.  2871  by  3.        Ans.    957. 

4.  7508  by  4. 

5.  6730  by  5. 

6.  6102  by  6. 

7.  3402  by  7. 


Ans.  1877. 
Ans.  1346. 
Ans.  1017. 


4ns.     486. 
8.  10024  by  8.      Ans.  1253. 

16.  2643--  3  =  ?    Ans.  881. 

17.  6235-^5  =  =  ?        Ans.  ? 


9.  95167  by  3.        Rem.  1. 

10.  12678  by  4.         Rem.  2. 

11.  75613  by  5.        Rem.  3. 


12.  14789  by  6. 

13.  95328  by  7. 

14.  18903  by  9. 

15.  74638  by  10. 

18.  4571--? 

19.  581-5=:? 


8" 


Rem.  5. 
Rem.  2. 
Rem.  3. 

Rem.  8. 

JSem.  1. 
Rem.  ? 


20>  How  many  yards  of  cloth  will  it  take,  at  5  cents 
a  yard,  to  amount  to  37295  cents?        Ans.  7459  yards. 

21.  A  merchant  spent  $33224  for  hats,  paying,  on  an 
average  $4  apiece;  how  many  hats  did  he  buy? 

Ans.  8306  hats. 

22.  A  farmer  received  $1950  for  a  lot  of  land  which 
he  sold   at   the   rate  of   $6  per  acre;    how   many   acres 
did  he  sell?  Ans.  ? 

23.  4308   chestnuts   were   divided   equally     among    6 
boys;  how  many  chestnuts  did  each  boy  receive?  Ans.  ? 


Division. 


115 


24.  There   are    12320   yards   in    7   miles ;    how   many 
yards  in  one  mile?  Ans.  1760. 

25.  I  counted  the   legs  of  all  the  horses   in  a  drove, 
and  found  that  there  were  476 ;  how  many  horses  were 
in  the  drove?  Ans.  ? 

LONG  DIVISION. 

136.  When  the   divisor  is  greater   than    12   we   write 
down  all  the  figures  employed,  and   call  the  operation 
Long  Division. 

137.  CASE  I. — When  the  quotient  is  not  greater  than  9. 
1.  Divide  91  by  21. 


EXPLANATION.  -  -  The  first  figure  of  the 
dividend  is  9,  and  that  of  the  divisor  is  2  ;  2 
in  9,  4  times.  Place  the  4  on  the  right,  mul- 
tiply it  by  21,  subtract  the  product  84  from 
91,  which  gives  7  remainder. 

2.  Divide  442  by  75. 

The  first  part  of  the  dividend  that  con- 
tains the  first  figure  (7)  of  the  divisor  is  44; 
7  in  44,  6  times.  Place  the  6  on  the  right, 
multipl  y  it  by  75,  and  since  the  result,  450, 
is  larger  than  442,  6  is  too  large.  Hence,  we 
take  the  next  less  number  (5),  put  it  in 
place  of  6,  multiply  it  by  75,  and  since  the 
result,  375,  is  less  than  442,  5  is  the  correct 
quotient,  and  the  answer  is  :  quo.  5  rein.  07. 

3.  Divide  337  by  35. 

Since  33  is  less  than  35,  we  say  3  in  33,  11 
times.  But  in  dividing  in  this  manner  we 
can  never  get  a  greater  quotient  than  9  ; 
hence,  instead  of  11  we  write  9  on  the  right, 
multiply  it  by  35,  and,  since  the  result,  315, 
is  less  than  337,  9  is  the  correct  quotient. 


OPERATION. 

21)91(4 
84 


OPERATION. 

75)442(6 
450 

75)442(5 
375 

67 


35)337(9 
315 

22 


116  Intermediate  Arithmetic. 

From  the  preceding  work  we  derive  the 

RULE. — I.  Divide  the  first  figure  or  figures  of  the  divi- 
dend by  the  first  figure  of  the  divisor ;  place  the  result  on 
the  right  and  call  it  the  trial  quotient. 

II.  Multiply  the  divisor  by  the  trial  quotient  and  place 
the  product  under  the  dividend;  if  it  is  larger  than  the 
dividend,  the  trial  quotient  is  too  large  and  must  be  dimin- 
ished; if  it  is  smaller,  subtract  it  from  the  dividend,  and 
if  the  remainder  is  less  than  the  divisor  the  work  is  correct  ; 
if  greater,  the  trial  quotient  is  too  small  and  must  be  in- 
creased. 

PROOF.  —  The  same  as  in  short  division. 
Divide: 

4. 
5. 
6. 
7. 
8. 
9. 

16. 
17. 

18.  Divide  517604  a  by  89325  a.  R&ni.  71069  a. 

138.  CASE  II. — When  the  quotient  is  more  than  9. 
1.  Divide  9156  by  21.  OPERATION. 

EXPLANATION.— 21  in  91,  by  CASE  I,  21)9156(436 

goes  4  times  and  rem.  7.     Place  4  on 

the  right  and  annex  5  'to  7,  making  75.  75 

By  CASE  I,  21  in  75,  3  times  and  rem.  gg 

12.     Place  3  on  the  right  and  annex  6  — — 
to  12,  and  we  find  by  CASE  I,  21  in  126 

goes  6  times,  which  place  on  the  right.  126 


117  by  23.  Ans.Q.5,R.2. 

10.  300  by  45.    Rem.  30, 

400  by  76.    Rem.  20. 

11.  967  by  98.    Rem.  85. 

311  by  88.    Rem.  47. 

12.  573  by  75.    Rem.  48. 

728  by  93.    Rem.  77. 

13.  805  by  237.   Rem.  94. 

643  by  75.    Quo.  8. 

14.  933  by  465.    Rem.  3. 

340  by  49.    Quo.  6. 

15.  1080  by  135.   Rem.  0. 

Divide  43657  by  8705.               Rem.  132. 

Divide  $34637  by  $9604.            Rem.  $5825. 

llivision.  117 


2.  Divide  876000  by  125. 


OPERATION. 


By  CASE   I,  125  in  87G,  7  times  and          125)876000(7008 
rem.  1.     Place  7  on  the  right  and  an-  £75 

nex  0,  the  next  figure  of  the  dividend 
to  1 ,  making  10.  Now  125  in  10,  0  times 
and  rem.  10.  Place  0  on  the  right  and 
annex  the  next  figure  0  to  10,  making 

100.  Now  125  in  100,  0  times  and  100  rem.  Place  0  on  the  right 
and  annex  the  next  figure  0  to  100,  making  1000.  By  CASE  I,  125 
in  1000,  8  times,  which  we  place  on  the  right. 

From  the  foregoing  examples  and  operations  we  de- 
rive the  following 

RULE. — I.  Write  the  divisor  on  the  left  of  the  dividend, 
with  a  line  between  them,  and  draw  a  line  on  the  right. 

II.  Find  how  many  times  the  divisor  is  contained  in  the 
least  number  of  the  left  hand  figures  of  the  dividend  that  will 
contain  it,  and  place  the  quotient  on  the  right. 

III.  Multiply  the  divisor  by  this  quotient  figure,  subtract 
the  product  from  the  figures  of  the  dividend  used,  and  to  the 
remainder  annex  the  next  figure  of  the  dividend. 

IV.  Divide  as  before,  and  continue  the  operation  until  all 
the  figures  of  the  dividend  have  be-en  brought  down. 

V.  When  one  of  the  partial  dividends  is  less  than  the  di- 
visor, write  0  for  the  next  figure  of  the  quotient,  and  bring 
down  the  next  figure  of  the  dividend. 

PROOF. — Add  the  remainder  to  the  product  of  the  divisor 
and  quotient;  the  result  should  be  equal  to  the  dividend. 

NOTE. — When  there  is  a  remainder  after  all  the  figures  of  the 
dividend  have  been  brought  down  and  divided,  it  may  either  be 
set  off  by  itself,  QT  it  may  be  written  over  the  divisor  and  annexed 
to  the  quotient. 


118 


Intermediate  Arithmetic. 


Divide : 


3.  625  by  25.  Ans.  25. 

4.  759  by  33.  Ans.  23. 

5.  864  by  36.  Ans.  24. 

6.  882  bv  42.  Ans.  21. 


8. 

9. 
10. 

11. 
12. 


1778  by  14.  Ans.  127. 
2169  by  18.  Rem.  9. 
3639  by  27.  Rem.  21. 
7540  by  59.  Rem.  47. 
35645  by  215.  Rem.  170. 
Quo.  138.  Rem.  ? 
Ans.  143fff. 


Ans. 
Ans. 

Ans.  4ff 
Ans. 


864  by  36. 
882  by  42. 
7.  270  by  18.   Ans.  15. 

13.  58650  by  425. 

14.  98629  by  687. 

15.  75863  by  3421. 

16.  10000  by  2749. 

17.  $132  by  $27. 

18.  $457  by  56. 

19.  How   many  hours  will  it  take  a  railway-train  to 
go  800  miles  at  the  rate  of  32  miles  an  hour? 

Ans.  25  hours. 

20.  What  is  the  weight  of  a  bale  of  cotton  if  25  bales 
weigh  11400  pounds?  Ans.  456  pounds. 

21.  How    many    hogsheads   of  sugar  will    it  take   to 
weigh  17400  pounds,  if  1  hogshead  weighs  1450  pounds? 

Ans.  12. 

22.  The   average   price   of  a  drove  of  horses   is   $127, 
and  the  price  of  the  whole  drove  is  $13335  ;  how  many 
horses  are  in  the  drove?  Ans.  105  horses. 

23.  AVith  $19608,    how  many  cows  can    I  buy  at  $43 
a  head?  Ans.  456. 

24.  How  many  bales,  each  weighing  475  pounds,  can 
be  made  of  93100  pounds  of  cotton  ?         Ans.  196  bales. 

25.  AVilliam   can   haul    1248   pebbles    in    his    wagon; 
how    many   trips    will    he   have   to    make  to    haul    off 
91104  pebbles?  Ans.  73. 

26.  If  the  distance  around  a  wheel  is  56  inches,  how 
many  times  will  the  wheel   turn   over   ir^   going  a  dis- 
tance of  7504  inches?  Ans.  134. 


Division.  119 

27.  At  what  price  per  head  must  I  sell  148  sheep  to 
receive  $1036?  Ans.  $7. 

28.  The  salary  of  the  President  of  the  United  States 
is  $50000  a  year;  how  much  is  that  a  day,  there  being 
365  days  in  1  year?  Ans.  8136fff 

29.  There  are  56  pounds   in   a  bushel  of  corn  ;   how 
many  bushels  in  12345  pounds?  Ans.  220ff. 

139.  Make  problems  of  the  following,  taking  the  terms 
in  order,  and  giving  the  answer  to  each  : 

30.  26992  h  H-  482  h  =  ? 

si.  3212  m  +  868  m  -+-  34  m  =? 

32.  4120  d--520  d-r-150  =  ? 

33.  703  p  X  8  —  204p-r-125  p==? 

CONTRACTIONS  IN  DIVISION. 

140.  CASE  I.—  When  the  divisor  is  10,  100,  1000,  etc. 
1.  Divide  1625  by  100. 

Dividing  according  to  the  rule  of  Long 

OPFR  \TION 

Division,  we  obtain  the  quotient  16  and 

remainder  25.  100)1625(16 

Now  we  observe  that  this  answer  could  100 

have  been  obtained  by  simply  cutting  off 
the  last  two  figures  (25)  of  the  dividend 
for  a  remainder,  and  taking  the  balance 


of  the  dividend  for  a  quotient. 

Hence,  to  divide  by  10,  100,  etc.,  we  have  the 

RULE.  —  Cut  off  from  the  right  of  the  dividend  as  many 
figures  as  tJiere  are  ciphers  at  the  right  of  the  divisor  ;  the 
remaining  figures  of  the  dividend  ivill  be  the  quotient,  and 
those  cut  off  on  the  right  will  be  the  remainder. 

2.  Divide  375  by  10.  Ans.  37,  Rem.  5. 


120  Intermediate  Arithmetic. 

3.  Divide  4316  by  100.  Ans.  43,  Rem.  16. 

4.  Divide  60524  by  1000.  Ans.  60,  Rcm.  524. 

5.  How  much  is  1  tenth  of  43  ?     1  hundredth  of  471  ? 

6.  There  are  10  dimes  in  one  dollar ;  how  many  dol- 
lars in  40  dimes?     260  dimes?    500  dimes? 

7.  A   farmer    having    $3254,    bought    horses    at    $100 
each ;   how   many  horses   did  he  buy,   and   how   many 
dollars  had  he  left? 

8.  A  dealer  has  1895  cigars,  and  wishes  to  put  them 
in  boxes  of  100  cigars  each  ;  how  many  boxes  does  he 
need,  and  how  many  cigars  will  he  have  left  over? 

141.  CASE  II. — When  the  divisor  is  any  number  with 
ciphers  annexed. 

1.  Divide  73153  by  2700. 

Dividing  by  the  rule  of  Long  Division,  QPERATION. 

we  obtain  the  quotient  27  and  remainder       27,00)731,53(27 
253,    which   result  could    have   been   ob-  ^A 

tained  thus : 

Cut  off  the  two  O's  of  the  divisor  and 
the  last  two  figures  of  the  dividend  ;  divide 
the  remaining  figures  of  the  dividend  253 

(731)  by  the  remaining  figures  of  the  di- 
visor (27),  and  to  the  remainder  (2)  annex  the  two  figures  cut  off 
(53)  for  the  true  remainder. 

Hence,  we  have  the 

RULE. — Cut  off  the  ciphers  from  the  divisor,  and  also  cut 
off  the  same  number  of  figures  from  the  right  of  the  dividend; 
divide  the  remaining  figures  of  the  dividend  by  the  remaining 
figures  of  the  divisor,  and  to  the  remainder,  if  any,  annex 
the  figures  cut  off'  from  the  dividend  for  a  true  remainder. 

2.  Divide  37657  by  50.  Ans.  753,  Rem.  7. 

3.  Divide  43787  by  600.  Ans.  72,  Rcm,  587. 

4.  Divide  35016  by  700.  Ana.  50,  Rem.  16. 


Division .  12 1 

5.  Divide  63242  by  3500.  Ana.  18,  R&m.  242. 

6.  Divide  71831  by  6400.  Ans. 

7.  Divide  93045  by  17000.  Ans.  5T 

8.  Divide  184973  by  23000.  Ans. 

9.  Divide  846  by  40.  Ans.  ? 

10.  Divide      7593  by  900.  Ans.  ? 

11.  Divide     23956  by  3700.  Ans.  ? 

12.  If  40  barrels  of  molasses   cost   $480,  what   is   the 
price  of  1  barrel?  •       Ans.  $12. 

13.  A  former  sold   600   acres  of  land   for  $7800;  how 
much  was  that  per  acre?  -4ns.  $13. 

14.  A  merchant  sold   8000  yards  of  cloth   for  184000 
cents  ;  how  much  was  that  per  yard  ?        Ans.  23  cents. 

15.  John  and  Henry  gather  6275  chincapins,  and  de- 
sire  to   put  them   in  sacks   containing   290   chincapins 
each ;  how   many  sacks   do   they  need,  and   how  many 
chincapins  will  be  left  over? 

MENTAL  EXERCISES. 

An  important  class  of  problems  involving  Multiplication 
and  Division. 

142.  i.  If  4  yards  of  cloth  cost  20  cents,  what  will  7 
yards  cost  at  the  same  rate?* 

OPERATION. 

EXPLANATION.— Since  4  yards  cost  20  cents  4)20 

we  divide  20  by  4  to  get  the  cost  of  1  yard,  — F — 

which  gives  5  cents.   Now,  since  1  yard  cost  _ 

5  cents,  we  multiply  5  by  7  to  get  the  cost  

of  7  yards.  35  cents. 

2.  If  5   yards  of   cloth   cost  40  cents,    what   will  8 
yards  cost?  Ans.   64  c. 

*The  words  "at  the  same  rate"  are  supposed  to  follow  several  of  the 
following  exercises. 


122  Intermediate  Arithmetic. 

3.  If  4  apples  cost  12  cents,  what  will  9  apples  cost? 

4.  If  6  peaches  cost  24  cents,  what  will  10  peaches 
cost?  Ans.  40  c. 

5.  If  7  melons  cost  70  cents,  what  will  9  melons  cost? 

6.  If  9  hats  cost  $27,  what  will  7  hats  cost?    Ans.  $21. 

7.  If   8  books    cost   $16,  what  will    11   books   cost? 

8.  If  3  tens  cost  $15,  what  will  7  tens  cost?  Ans.  $35. 

9.  If  2  threes  cost  $24,  what  will  5  threes  cost? 

10.  If  5  fourths  cost   S30,  what   will   3  fourths  cost? 

11.  If  4  11  cost  24  cents,  what  will  9  u  cost?   Ans.  ? 

12.  If  4  boys  kill  8  squirrels,  how  many  will  5  boys 
kill?  *  .  Ana.    ? 

13.  If  5    cats   catch    15    rats,  how  many  will   7   cats 
catch  ?  Ans.    ? 

14.  If  6  boys  eat  18  biscuits,  how  many  will  12  boys 
eat?  Ans.   ? 

15.  If  5  girls  have  40  fingers,  how  many  fingers  have 
7  girls?  Ans.   ? 

16.  If  7  horses  have  28  feet,  how  many  feet  have  12 
horses  ?  Ans.    ? 

17.  If  5  gallons  =  =20  quarts,  then  8  gallons  =  ? 

Ans.  32  quarts. 

18.  If  4  quarts  =  8  pints,  than  11  quarts  =? 

19.  If  6  dimes  =  60  cents,  then  13  dimes  =  ? 

20.  If  $9  =  90  dimes,  then  $6  =  ?          Ans.  60  dimes. 

WRITTEN  EXERCISES. 

1.  If  18  acres  cost  $270,  what  will  20  acres  cost? 

Ans.  $300. 

2.  If  25  tables  cost  $400,  what  will  12  tables  cost? 

Ans.  $192. 

3.  If  31  cows  cost  $465.  what  will  60  cows  cost? 

Ans.  $900. 


Division.  123 

4.  If  43  readers  cost  688  cents,  what   will  25  readers 
cost?  Ans.  400  cents. 

5.  If  752  sheep  cost  $4512,  what  will  137  sheep  cost? 

Ans.  $822. 

6.  If  20  bushels=640  quarts,  then  17  buxhels  -  =  ? 

Ans.  544  quarts. 

7.  If  15  yards  =  540  inches,  then  23  yards  =  ? 

-4ns.  828  inches. 

8.  If  75  ones  =  675  ninths,  then  63  ones  =  ? 

Ans.  567  ninths. 

143.  PARALLEL  PROBLEMS. 

1.™  What  number  multiplied  by  12  will  make  108? 

2.  One  of  the  factors  of  4375  is  175,  what  is  the  c 
factor?  Ans.  25. 

3.m  If  9  boys  together  catch  72  fishes,  how  many  does 
each  boy  catch  on  an  average? 

4.  A  field  of  57  acres  produced  1539  bushels  of 
wheat,  what  was  the  average  product  of  an  acre? 

Ans.  27  bushels. 

5.™  If  11  vards  of  cloth  cost  132  cents,  what  is  the 

%' 

cost  of  1  yard? 

6.  A  man  sold  59  acres  of  land  for  $1062 ;  what 
did  he  receive  per  acre? 

7.™  What  number  is  contained  in  68  9  times  and  5 
over? 

8.  A  man  had  $937,  and,  after  paying  some  labor- 
ers at  the  rate  of  $25  apiece,  had  812  left;  how  many 
laborers  were  there?  Ans.  37. 

9.m  How  often  is  13  a  less  5  a  contained  in  56  a? 
10.     A  clerk's  yearly  salary  is  $1500  and  his  expenses 
8945;  in  how  many  years  can  he  lay  up  $4440? 

Ans.  8  years. 


124  Intermediate  Arithmetic. 

ll.m  A  boy  sold  a  merchant  4  oranges  at  6  cents 
apiece,  and  9  apples  at  4  cents  apiece,  and  took  his 
pay  in  cigars  at  5  cents  a  piece ;  how  many  cigars  did 
he  get?  Ans.  12  cigars. 

12.  A  farmer  sold  27  cords  of  wood  at  $5  a  cord, 
and  47  hogs  at  $7  apiece,  and  took  in  exchange  flour 
at  $8  a  barrel;  how  many  barrels  did  he  get? 

Ans.  58  barrels. 

13.™  How  many  minutes  will  it  take  2  boys  to  re- 
move 80  rails  if  one  boy  removes  6  rails,  and  the  other 
4  rails,  every  minute? 

14.  Two  railway  trains  are  540  miles  apart,  and  travel 
towards  each  other  at  the  rates  of  25  miles  and  20 
miles  per  hour ;  how  many  hours  before  they  will 
meet?  Ans.  12  hours. 

15.™  If  a  horse  walks  35  miles  in  7  hours,  how  far 
will  he  walk  in  11  hours  at  the  same  rate? 

16.  If  a  railway  train  goes  304  miles  in  16  hours, 
how  far  will  it  travel  in  21  hours  at  the  same  rate? 

Ans.  399  miles. 

17.™  How  much  is  ^  of  the  sum  of  12  and  8? 

18.  What  is  the  average  width  of  a  field  which  is 
332  yards  wide  at  one  end,  and  478  yards  wide  at  the 
other?  Ans.  405  yards. 

19.™  How  much  is  ^  of  the  sum  of  11,  9,  and  4? 

20.  A  farmer  killed  3  hogs ;  one  weighed  165  pounds, 
another  173  pounds,  and  the  third  181  pounds;  what 
was  the  average  weight  of  the  three  hogs? 

Ans.  173  pounds. 

21. m  Henry  started  on  a  journey  of  62  miles,  and 
traveled  at  the  rate  of  5  miles  a  day  for  4  days ;  how 
many  days  will  it  take  him  to  complete  the  journey 
if  he  goes  at  the  rate  of  6  miles  a  day? 

22.     A  laborer  engaged   to  remove  4703  bricks.     After 


Division-.  125 

working  24  hours,  removing  125  bricks  each  hour,  how 
many  hours  will  it  take  him  to  complete  the  work  if 
he  removes  131  bricks  per  hour?  Ans.  13. 

23. m  Frank  sold  10  oranges  at  6  cents  apiece,  and  with 
the  money  bought  apples  at  5  cents  apiece  ;  how  many 
apples  did  he  get? 

24.  A  man  sold  144  horses  at  $135  apiece,  and  in- 
vested the  money  in  land  at  $18  per  acre;  how  many 
acres  did  he  get?  Ans.  1080  acres. 

25.™  A  boy  spent  42  cents  for  cakes,  and  1-sixth  as 
much  for  nuts ;  what  are  the  c  parts  of  what  he  spent 
in  all?  How  much  did  he  spend? 

26.  A  farmer  has  306  acres  in  one  field,  and  1-eigh- 
teenth  as  much  in  another  field  ;  how  much  has  he  in 
both  fields?  Ans.  323  acres. 

144.  QUESTIONS  FOR  REVIEW. 

What  is:  1.  Division?  2.  The  dividend?  3.  The  divisor?  4. 
The  qnotient  ?  5.  The  sign  of  division  ? 

What  is  denoted  by  the  sign  -=-  ? 

Name  the  three  principles  of  division.  Can  division  be  ef- 
fected by  subtraction? 

What  is  the  relation  of  division  to  multiplication?  What  do 
we  find  by  division  ? 

What  is  meant  by  :  1.  Short  division?  2.  Long  division?  Give 
the  rule  for  each. 

How  do  we  find:  1.  The  half  of  a  number?  2.  The  third  of  a 
number?  3.  The  fourth f  4.  The  fifth?  etc. 


DIVISORS  AND  MULTIPLES. 


145.  A  Divisor  of  a  number  is  one  of  its  factors.* 

Thus,  4  is  a  divisor  of  12,  since  4X  3=  12.  Also,  1,  2,  3,  4,  6, 
and  12  are  divisors  of  12,  since  each  is  contained  in  12  an  exact 
number  of  times. 

1°.  1  is  a  divisor  of  every  number. 
2°.  Every  number  is  a  divisor  of  itself. 

Any  number  is  exactly  divisible  : 

3°.  By  2,  if  its  last  figure  is  divisible  by  2. 

4°.  By  4,  if  its  last  two  figures  are  divisible  by  4. 

5°.  By  8,  if  its  last   three  figures  are  divisible  by  8. 

6°.  By  3  or  9,  if  the  sum  of  its  figures  is  divisible 
by  3  or  9. 

7°.  By  5,  if  it  ends  in  5  or  0. 

8°.  By  6,  if  it  is  divisible  by  2  and  3. 

MENTAL  EXERCISES. 

How  many  times  is : 

1.  1  contained  in  11?     1  contained  in  143? 

2.  13  contained  in  13?     97  contained  in  97? 
Which  of  the  numbers  2,  3,  4,  5,  6,  8,  9,  and  10  are 

divisors  of :  250?     3056?     4581?     1722?    45460?     761? 
23202?     45128?     37301?     371820? 

"The  terms  numbers,    divisors,   multiples,  and  factors,  as  here  used,  de- 
note integers. 
(126) 


Divisors  and  Multiples.  127 

146.  A   Common   Divisor  of  two  or   more   numbers  is 
any  number  which  will  exactly  divide  all  of  them. 

Thus,  2  and  4  are  common  divisors  of  8  and  12,  since  they  di- 
vide each  of  them  without  a  remainder. 

EXERCISES. 

1.  What  are  the  common  divisors  of  6,  12,  and  24? 

Ans.  2,  3,  and  6. 

2.  What  are  the  common  divisors  of  8,  16,  and   24? 

Aiis.  2,  4,  and  8. 

3.  What  are  the  common  divisors  of  12,  18,  and  24? 

Ans.  2,  3,  and  6. 

4.  What  are  the  common  divisors  of  36,  54,  and  72  ? 

Ans.  2,  3,  6,  9,  and  18. 

5.  What  are  the  common  divisors  of  24,  36,  and  48? 

Ans.  2,  3,  4,  6,  and  12. 

147.  The  Greatest    Common   Divisor   of  two   or   more 
numbers,  denoted   by  G.  C.  D.,  is   the   greatest   number 
that  will  exactly  divide  each  of  them. 

Answer  these  five  questions  by  referring  to  the  pre- 
ceding exercises :  What  is  the  G.  C.  D.  of  6,  12,  24  ? 
Of  8,  16,  24?  Of  12,  18,  24?  Of  36,  54,  72?  Of  24, 
36,  48? 

148.  PRINCIPLE. — The    c   factors    of  the   G.  C.  D.    of 
several  numbers  are  all  the  factors  common  to  all. 

Thus,  the  c  factors  of  12  are  2,  2,  3, 
the  c  factors  of  18  are  2,  3,  3. 

Now,  from  each  we  may  take  the  factor  2,  then  the  factor  3, 
and  as  no  equal  factors  remain,  2  and  3  are  the  c  factors  of  the 
G.  C.  D.  of  12  and  18. 


128  Intermediate  Arithmetic. 

149.  What  is  the  G.  C.  D.  of  18,  27,  and  36? 

EXPLANATION. — First  divide  18,  27,  and  OPERATION. 

36  by  any  number  that  will  divide  each  of         3)18    27.   36. 
them,  as  3.    Then  divide  the  quotients  6,         o\~c Q    7i7 
9,  and  12,  by  any  number  that  will  exactly 
divide  each  of  them,  as  3.     Now  there  is  2.     3.     4. 

no  number  except  1  that  will  divide  the 

quotients  2,  3,  4.     Hence,  3  and  3,  the  two  divisors,  are  the  c  fac- 
tors of  the  G.  C.  D. ;  that  is,  the  G.  C.  D.  is  9. 


WRITTEN  EXERCISES. 

1.  What  is  the  G.  C.  D.  of  12  and  30?  Am.     6. 

2.  What  is  the  G.  C.  D.  of  24,  30,  and  54  ?         Ans.     6. 

3.  What  is  the  G.  C.  D.  of  35,  56,  and  70?         Ans.    7. 

4.  What  is  the  G.  C.  D.  of  18,  36,  and  72?         Ans.  18. 

5.  What  is  the  G.  C.  D.  of  48,  72,  and  144?       Ans.  24. 

6.  What  is  the  G.  C.  D.  of  32,  48,  64,  and  160?  Ans.  16. 

7.  What  is  the  greatest  common  divisor  of  60,  90,  150, 
and  210?  Ans.  30. 

150.  A  Prime  Number  is  a  number  which  has  no  di- 
visors except  itself  and  1. 

Thus,  1,  2,  3,  5,  7,  etc ,  are  prime  numbers. 

151,  A    Composite    Number   is    a    number   which   has 
other  divisors  than  itself  and  1. 

Thus,  4,  6,  8,  9,  10,  etc.,  are  composite  numbers. 

Is  13  a  prime  or  a  composite  number? 

Ans.  A  prime  number,  as  it  has  no  divisors  except 
1  and  13. 

Is  21  a  prime  or  a  composite  number? 

Ans.  A  composite  number,  as  it  can  be  divided  by  3 
and  also  by  7. 


Divisors  and  Multiples.  129 

152.  Prime  Factors  are  factors  which  are  prime  num- 
bers. 

Thus,  2,  2,  and  3  are  the  prime  factors  of  12. 

WRITTEN  EXERCISES. 

153.  1.  Write  all   the  prime   numbers  between  1   and 
25 ;  25  and  50  ;  50  and  75 ;  75  and  100. 

2.  What  are  the  prime  factors  of  56?          OPERATION. 

EXPLANATION.  — We  divide  the  number  by  any 
prime  factor ;    then  divide   the  quotient  by  any  2)28 

prime  factor,  etc.,  until  the  quotient  1  is  obtained.  2)14 

The   several   divisors   are   the   prime   factors   re- 


LSJL  J..U-1*^         ldV>  L*yi  O        1  t-  —.  -_ 

quired.     Hence,  the  answer  is  2,  2,  2  and  7.  ^    ' 

What  are  the  prime  factors  of: 


3.  50?  Ans.  2,  5.  5. 

4.  60  ?  Ans.  2,  2,  3,  5. 

5.  108?  Ans.  2,  2,  3,  3,  2. 

6.  640?  Ans.  5,  seven  2's. 

7.  455  ?  Ans.  5,  7,  13. 


8.  680?  Ans.  ? 

9.  1155?  Ans.  ? 

10.  7800?  Ans.  ? 

11.  2310?  Ans.  ? 

12.  4290?  Ans.  ? 


154.  A  Multiple  of  a  number  is  a  number  which  con- 
tains it  an  exact  number  of  times. 

Thus,  24  contains  6  exactly  4  times,  hence  24  is  a  multiple  of  0. 

Is  16  a  multiple  of  8?  25  a  multiple  of  5?  27  of  7? 
36  of  9?  42  of  6?  54  of  9?  63  of  7?  64  of  10?  72 
of  12? 

155.  A  Common  Multiple  of  two  or  more   numbers  is 
a  number  which  is  exactly  divisible   by  each  of  them. 

Thus,  12  is  a  common  multiple  of  2,  3,  4,  and  0 ;  18  of  2,  3,  6, 

and  9. 

N.  i.— 9. 


130  Intermediate  Arithmetic. 

EXERCISES. 

1.  Name  three  common  multiples  of  2,  3,  and  6. 

Am.  6,  12,  and  18. 

2.  Name  three  common  multiples  of  4,  5,  and  10. 

Ans.  20,  40,  60. 

3.  Name  three  common  multiples  of  3,  4,  12. 

Ans.  12,  24,  36. 

4.  Name  three  common  multiples  of  7  and  2;  5  and 
8;  3,  7,  and  2;  5  and  12;  3,  5,  and  10;  2,  4,  and  8. 

156.  The  Least  Common  Multiple  of  two  or  more  num- 
bers, denoted  by  L.  C.  M.,  is  the  smallest  number  which 
these  numbers  will  exactly  divide. 

Answer  these  questions  by  referring  to  the  preceding 
exercises:  What  is  the  L.  C.  M.  of  2,  3,  and  6?  Of  4, 
5,  and  10?  Of  3,  4,  and  12? 

What  is  the  L.  C.  M.  of  5  and  6?  6  and  7?  6  and 
8?  5  and  9?  7  and  10?  2,  3,  and  5? 

157.  PRINCIPLE.— The  c  factors  of  the  L.  C.  M.  of  two 

or  more  numbers  are  all  the  prime  factors  of  each. 

Thus,  the  prime  factors  of  12  are  2,  2,  and  3. 

30  are  2,  3,  and  5. 
"  "  50  are  2,  5,  and  5. 

Taking  out  the  facto.  2  from  each,  then  the  factor  3  from  the 
first  and  second,  then  the  factor  5  from  the  second  and  third, 
there  are  left,  2  in  the  first,  and  5  in  the  third.  Hence,  the 
c  factors  of  the  L.  C.  M.  of  12,  30,  and  50  are  2,  3,  5,  2,  and  5. 

WRITTEN  EXERCISES. 

158.  What  is  the  L.  C.  M.  of  12,  15,  and  25? 

EXPLANATION.  —  Divide  any  two  or  more  of  the  numbers  by 
any  common  prime  factor,  as  3,  and  bring  down  with  the  quo- 


Divisors  and  Multiples.  131 

tients  4  and  5  such  numbers  (25)  as  do  not  OPERATION. 

contain  the  divisor.     Again,  divide  out  by  5,         o\io    15    or 
as  it  is  a  prime  factor  common  to  5  and  25, 
and  bring  down   the  4.     Now  there  is  no        5)   4      5    25 
factor  except  1  that  will  divide  two  of  the  415 

numbers  4,  1,  and  5.     Hence,  the  two  di- 
visors 3  and  5,  and  the  quotients  4  and  5  are  the  c  factors  of 
the  L.  C.  M. ;  that  is,  the  L.  C.  M.  is  3  X  5  X  4  X  5  =  =  300,  as  it 
contains  all  the  prime  factors  of  12,  15,  and  25. 

Find  the  L.  C.  M.  of: 

2.  6  and  12  ;  9  and  15  ;  20  and  25.        Ans.  12 ;  45  ;  100. 

3.  12  and  20 ;  16  and  24 ;  35  and  42.    Ans.  60;  48 ;  210. 

4.  2,  4,  5,  and  12 ;  3,  7,  9,  and  14.  Ans.  60;  126. 

5.  5,  8,  10,  and  12;  7,  9,  12,  and  18.          Ans.  120 ;  252. 

6.  5,  6,  10,  and  15;  6,  12,  15,  and  20.  Ans.  30;  60. 

7.  15,  20,  30,  and  40;  16,  20,  32,  and  40.  Ans.  120;  160. 

8.  25,  36,  50,  and  72;  48,  60,  96,  and  120. 

Ans.  1800 ;  480. 

9.  5,  7,  11,  and  15  ;  3,  7,  13,  and  39.       Ans.  1155 ;  273. 
10.  4,  5,  6,  10,  12,  15,  20,  and  30.  Ans.  60. 

159.  QUESTIONS  FOR  REVIEW. 

"What  is:  1.  A  divisor  of  a  number?  2.  A  common  divisor  of 
two  or  more  numbers?  3.  The  G.  C.  D.  of  two  or  more  numbers? 

Give  an  example  of  each. 

What  is:  1.  A  prime  number?  2.  A  composite  number ?  3. 
A  prime  factor?  4.  A  multiple  of  a  number?  5.  A  common  mul- 
tiple of  two  or  more  numbers  ?  6.  The  L.  C.  M.  of  two  or  more 
numbers?  Give  an  example  of  each. 

What  is  the  principle  of :  1.  The  G.  C.  D.  ?    2.  The  L.  C.  M.  ? 

When  is  a  number  exactly  divisible  by :  2  ?  3?  4?  5?  G? 
8?  9?  10? 


COMMON  FRACTIONS. 


INDUCTIVE  EXERCISES. 

160.  When  an  apple,  an  orange,  a  number,  or  a  bar 
of  soap  is  divided   into   two  equal  parts,  what  is  each 

part  called?     Hew  is  1-half 

written?  Ans.  %.    W 

How    do    we   get   ^    of  a 
thing  ? 

Ans.  Divide  it  into  two  equal  parts,  and  take  one  of 
the  parts. 

What  is  i  of  a  pile  of  2  books  ?    £  of  82  ?    J  of  $4  ? 
i  of  $16? 

When   an   apple,  an   orange,  a   number,  or  a  bar  of 

soap  is    divided   into   three    ^ ^_     ^ 

equal    parts,    what    is    each 

part  called?     What  are  two 

of  the   parts    called?      How   are    1-third    and    2-thirds 

written?  Ans.  %  and  -f. 

How  do  we  get  -|-  of  a  thing  ? 

Ans.  Divide  it   into   three  equal  parts  and   take  one 
of  the  parts. 

What  is  i  of  a  pile  of  3  books  ?    \  of  $3  ?    i  of  $6  ? 
1  of  $12?    i  of  24c.?     \  of  60  bushels? 

o  o  o 

How  do  you  get  f  of  a  thing? 

Ans.  Divide  it  into  three   equal   parts  and   take  one 
of  the  parts  2  times. 

(132) 


Common  Fractions.  133 

What  is  1  of  a  pile  of  3  books?    |  of  $3?    J  of  $12? 
|  of  27c.? 

When  an  apple,  an  orange,  a  number  or  a  bar  of  soap 
is   divided    into   four   equal 
parts,    what    is    each    part 
called?      What    are    two    of 

the  parts  called?     What  are  three  of  the  parts  called? 
How  are  1-fourth,  2-fourths,  and  3-fourths  written? 

Ans.  J,  f ,  and  f . 

How  do  we  get  ^  of  a  thing? 

Ans.  Divide  it  into  four  equal  parts,  and  take  one  of 
the  parts. 

What  is  \  of  a  pile  of  4  books?    J  of  $4?    J  of  $20? 
\  of  40c.  ? 

How  do  we  get  f  of  a  thing  ? 

Ans.  Divide  it  into  four  equal  parts  and  take  1  part 

2  times. 

How  much  is  |  of  a  pile  of  4  books  ?     }  ot  4  ?    f  of 
16?    |  of  32? 

How  do  we  get  f  of  a  thing? 
Ans.  Divide  it  into  four  equal  parts,  and  take  1  part 

3  times. 

How  much  is  f  of  a  pile  of  4  books  ?    j  of  4  ?    f  of 


12?    f  of  40? 


DEFINITIONS. 


161.  Fractional  Parts  are  parts  obtained  by  dividing 
any  thing,  or  a  unit,  into  any  number  of  equal  parts. 

Thus :  halves,  thirds,  fourths,  fifths,  sevenths,  tenths,  etc.,  are 
fractional  parts. 

162.  A  Fractional  Unit  is  one  of  the  equal  fractional 
parts  into  which  a  thing  is  divided. 


134  Intermediate  Arithmetic. 

Thus-  I  1'half'  !-third,  1-fourth,  1-fifth,  1-tenth,  1-twelfth,  j 

ill  \  \  TO  T5  > 

are  fractional  units. 

163.  A  Fraction  is  a  fractional  unit  taken  one  or  more 

times. 

C  l-fourtb,  2-fifths,  3-sevenths,  7-tenths,  ~\ 

Thus  :]       1  X  i       2  X  i          3  X  \         7  X  TV     [  are  fractions. 

11  371 

V.  t  3  7  T7 

164.  The  Terms  of  a  fraction  are  the  two  numbers  used 
to  express  it. 

165.  The  Denominator   is  that  term  which   names  the 
parts  expressed  by  the  fraction.     It  is  written  below  the 
horizontal  line. 

166.  The  Numerator  is  that  term  which  numbers  the 
parts  expressed  by  the  fraction.    It  is  written  above  the 
horizontal  line. 

Thus :  the  terms  of  the  fraction  £  are  5  and  6 ;  the  denomina- 
tor is  6,  and  the  numerator  is  5. 

The  fraction  §  means : 

1°.  5  of  the  parts  when  1  unit  has  been  divided  into  6  equal 
parts. 

2°.  The  fractional  unit  1-sixth,  or  £,  taken  5  times. 

3°.  The  quotient  of  5  divided  by  6. 

Read  each  of  the  following  fractions,  name  the  terms, 
the  numerator,  the  denominator,  the  fractional  unit, 
the  number  of  fractional  units,  and  give  the  three 
meanings  of  each  fraction : 

I;       I;       £;       A;       £j       ¥;       5  ninths;       7  twelfths. 

167.  A  Proper  Fraction  is  one  of  which  the  numera- 
tor is  less  than  the  denominator ;  as  -f-,  f ,  f ,  etc. 

168.  An  Improper  Fraction  is  one  of  which  the  numer- 
ator equals  or  exceeds  the  denominator ;  as  f ,  \°-,  ^,  etc. 


Common  Fractions.  135 

169.  A  Mixed  Number  is  an  expression  consisting  of 
a  whole  number  and  a  fraction  ;  as  2|,  5J,  12  J,  etc. 

170.  The  Value  of  a  fraction   is   the  quotient  of  the 
numerator  divided  by  the   denominator. 

The  value  of  f  is  2  ;  of  -1/  is  4  ;  of  *f-  is  5. 

When  the  numerator  equals  the  denominator  the  value  of  the 
fraction  equals  1  ;  as  f,  f  ,  ||,  etc. 

The  value  of  a  proper  fraction  is  less  than  1,  as  its  numerator  is 
less  than  its  denominator. 

The  value  of  an  improper  fraction  is  equal  to  or  exceeds  1,  as 
its  numerator  equals  or  exceeds  its  denominator. 

171.  A  Compound  Fraction  is  a  fraction  of  a  fraction  ; 
as  i  of  i  |  of  f  of  T5T. 

MENTAL  EXERCISES. 


172.   l.  How  many  halves  (£)  in  one  (1)  ?     Why? 

2.  What  is  the  wrorth  of  one  orange,  if  1  half  of  it  is 
worth  5  cents  ?     6c.  ?     lOc.  ? 

3.  How  many  thirds  (-J)  in  one  (1)  ?     Why  ? 

4.  What  is  the  worth  of  one  bale  of  cotton,  if  1  third 
of  it  is  worth  $10?     812?    $20? 

5.  How  m^nj  fifths  (i)  in  one  (1)  ?     Why? 

6.  What  is  the  length  of  a  pole,  if  1  fifth  of  it  is  3 
feet  long?     7  feet  long?     10  feet? 

7.  How  many  sevenths  (-J-)  in  one  (1)  ?     Why  ? 

8.  What   is   the  weight  of  a  rock   if  1  seventh  of  it 
weighs  5  pounds?     9  pounds? 

9.  How  many  tenths  in  one  (1)  ?     Why  ? 

10.  How  many  marbles  in  a  box,  if  1  tenth  of  them  is 
6  marbles?     9  marbles?     25  marbles? 

11.  What   is  the  number   whose   half  is   1  ?     Whose 
third  is  2  ?      Whose  fourth   is  5  ?      Whose   sixth  is  2  ? 
Whose  ninth  is  3?     Whose  tenth  is  7? 


136 


Intermediate  Arithmetic. 


12.  How  much  is  ^-fourths  (f)  of  28? 

ANALYSIS.— I  fourth  of  28  is  7  ;  3  fourths  of  28  is  3  times  7  =  =  21,  Ans. 

13.  How  much  is  2  ^refe  of  12?     f  of  15?    f  of  30? 

14.  How  much  is  3  fourths  of  16?    f  of  20?    f  of  40? 

15.  How  much  is  4  fifths  of  20?     f  of  30?    f  of  $50? 

16.  How  much  is  5  sixths  of  18? 

17.  How  much  is  7  ninths  of  18? 

18.  How  much  is  5  twelfths  of  24?     |of36? 

19.  If  one  acre  of  land  cost  $12,  what  is  the  cost  of 


fof24?    |of60c.? 
f  of  36?    |  of  45c.? 


of  an  acre  ?     -J-  of  an  acre  ?     f  of  an  acre  ? 

20.  If  one  bushel  of  potatoes  cost  30  cents,  what  is  the 
cost  of  -J-  of  a  bushel  ?    f  of  a  bushel  ?    ^  of  a  bushel  ? 

21.  5  sixths  of  a  number  is  10,  what  is  the  number? 

ANALYSIS.— If  5  sixths  is  10,  I  sixth  is  $  of  10,  or  2 ;  hence,  the 
number  is  6  times  2  -=  12,  Ans. 

22.  What  is  the  number  of  which  5  sixths  is  20?    f  is 
20?    4-  is  16?    f  is  10?    A  is  21? 

(  O  1  A 

23.  What  will  a  melon  cost  if  f  of  it  cost  20  cents? 
If  |  of  it  cost  18  cents?    If  f  of  it  cost  35  cents? 

FUNDAMENTAL  PRINCIPLES. 

173.  CASE  I. — To  multiply  and  divide  fractional  units 

by  whole  numbers. 

Into   how    many    parts   is 

this  bar  of  soap  divided  ? 
What  is  1  part  called?  Is 
it  a  fractional  unit? 

If  each  of  the  three  parts 
is  cut  into  two  parts,  how 
many  parts  will  there  be  in 
all?  What  is  1  part  called? 
Is  it  a  fractional  unit 


Common  Fractions. 


137 


Is  1  part  of  the  first  bar  equal  to  2  times  1  part  of 
the  second  bar? 

What  does  this  show  ?          Ans.  That  2  times  ^  =-  J. 
What  else  does  it  show  ?     Ans.  That  1  half  of  ^  =  1. 

Hence, 

1°.  To  multiply  a  fractional  unit  by  a  number,  we  may 
divide  the  denominator  by  that  number. 

2°.  To  divide  a  fractional  unit  by  a  number,  we  multiply 
the  denominator  by  that  number. 

MENTAL  EXERCISES. 

How  much  is  : 

1.  2  times  £?    Ans. 

2.  4  times  £  ?    Ans. 

3.  5  times  y1^?  Ans. 

How  much  is : 

10.  1  half  of  i? 

11.  1  fourth  of  £' 

12.  1  s*atf^  of  J? 

13.  1  fifth  of  4-? 


4' 

4, 

2 

X 

Jl 

12 

? 

"2- 

5 

7 

X 

2T 

? 

¥• 

6 

11 

X 

filf 

? 

75X1 

8.  8)  ;  \  fortieth  f 

9  3  X  1  fifteenth? 


-4ns. 


Ans. 


14. 
15. 
16 

17. 


1  tenth  -^ 
1  third  -±- 
1  ninth  -r- 


174.  CASE  II. — To  change  the  fractional  unit. 

Into  how  many  equal  parts 
is  this  bar  of  soap  divided? 
What  is  1  part  called?  2 
parts  ?  Is  J  a  fractional  unit? 

Here  is  an  equal  bar ;  in- 
to how  many  parts  is  it 
divided?  What  is  1  part 
called  ?  2  parts  ?  etc.  Is  £ 
a  fractional  unit? 

Is  1  part  of  the  first  bar,  or 
the  second,  or  f  ? 


equal  to  two  parts  of 


138  Intermediate  Arithmetic. 

Are  2  parts  of  the  first  bar  equal  to  4  parts  of  the 
second  ? 

What  does  this  show  ?       Ans.  That  f  =  =  f,  and  $==-§. 

Hence, 

1°.  Multiplying  both  terms  of  a  fraction  by  the  same 
number  does  not  alter  the  value  of  the  fraction. 

2°.  Dividing  both  terms  of  a  fraction  by  the  same  num- 
ber does  not  alter  the  value  of  the  fraction. 

EXPLANATION.— Can  24  be  written  thus:  6  fours  f 
Thus:  12  twos?  Are  6  fours  equal  to  12  twos?  Why? 
Ans.  The  unit  two  is  half  of  the  unit  four,  but  is  taken 
twice  as  often,  since  12  is  twice  6;  hence,  they  are  equal. 

Are  3  fifths,  or  f ,  equal  to  6  tenths,  or  T%  ?     Why  ? 
Ans.  The  unit   1   tenth  is  half  of  the  unit  1  fifth,  but  is 
taken  twice  as  often,  since  6  is  twice  3 ;  hence,  f  —  -f^. 

Why,  then,  is  the  value  of  a  fraction  not  changed  by 
multiplying  both  terms  by  the  same  number  ?  Ans. 
Because  it  decreases  the  fractional  unit  in  the  same 
ratio  that  it  increases  the  number  of  times  it  is  taken. 

Why  is  the  value  of  a  fraction  not  changed  by  divid- 
ing both  terms  by  the  same  number?  Ans.  Because  it 
increases  the  fractional  unit  in  the  same  ratio  that  it 
decreases  the  number  of  times  it  is  taken. 

REDUCTION  OF  FRACTIONS. 

175.  Reduction  of  a  Fraction  consists  in  changing  its 

o       o 

terms  without  altering  its  value. 

176.  CASE  I. — To  reduce  a  fraction  to  its  lowest  terms. 

A  fraction  is  in  the  lowest  terms  when  no  number 
greater  than  1  will  exactly  divide  its  numerator  and 
denominator. 


Common  Fractions.  139 

1.  Reduce  f£  to  its  lowest  terms.  OPERATION. 

f)\  3  0  15 

)  4"9"  ~  "  ^TF 

EXPLANATION.— Dividing  both  terras  of 
f-$  by  2,  we  get  £-$.   Now  dividing  both  terms  5)13.  —  3 

of  Jo-  by  5,  we  obtain  £. 

In  the  second  operation  we  divide  both        2d  OPERATION. 
terms  by  their  G.  C.  D.,  10.  10)  ^  = 

Hence  the, 

RULE. — Divide  both  terms  of  the  fraction  by  any  number 
greater  than  1  that  will  exactly  divide  them,  and  continue 
the  operation  as  long  as  possible. 

Or,  Divide  both  terms  by  their  G.  C.  D. 

WRITTEN  EXERCISES. 

Reduce  to  lowest  terms: 

ol4212445        75  A^o        232        9        9 

*•    To  i  "5  "6  5    T65    635     1  2"o'  -«./«*.       -5,     g-,     3,       I 

Q        84        112        81        6313 5  AW*        343        9        9 

P1    T9"65    T4~"05    TO^S"?    "815    TTO'  /1/Wf.       y,     5-,    4^, 

A.      2_0_7      j_2_8        39        J.9.A     _5  2_8_  A^Q     J.3.       ?      3        9      1_2. 

*•      99"5      885     1435     2255     17712'  t<5<     115  11?     '      23* 

177.  CASE  II. — To  reduce  a  whole  number  to  a  frac- 
tion. 

1.  Reduce  4  to  a  fraction  whose  denominator  is  3. 

EXPLANATION.— We  write  4  thus:  f,  then  OPERATION. 

multiply  both  terms  by  3,  and  obtain  -1-/-.  r-L2-. 

Hence,  the 

RULE. — Multiply  the  whole  number  by  the  given  denom- 
inator and  place  the  product  over  the  denominator. 

WRITTEN  EXERCISES. 

Reduce  : 

2.  5  to  a  fraction  whose  denominator  is  3.     Ans.  ^-. 

3.  7  to  a  fraction  whose  denominator  is  8.     Ans. 


140  Intermediate  Arithmetic. 

4.  9  to  a  fraction  whose  denominator  is  6.      Ans.  ^-. 

5.  11  to  a  fraction  whose  denominator  is  5.    Ans.     ? 


Reduce  : 

6.  7  to  fifths.  Ans.  -3/. 

7    8  to  thirds.          Ans.  -¥-. 


8.  15  to  fourths.         Ans.   ? 

9.  20  to  tenths.  Ans.    ? 


178.  CASE  III. — To  reduce  a  fraction  to  higher  terms. 

1.  Reduce  &  to  a  fraction  whose  denominator  is  15. 

o 

EXPLANATION. -Write  f,  and  on  its  right 
place  15  with  a  line  above  it.    Now  say  5  in  OPERATION. 

15  3  times.    3X3  =  9,  which  place  over  the  f  =  =73- 

15. 

Hence,  the 

RULE. — Divide  the  denominator  of  the  required  fraction 
by  the  denominator  of  the  given  fraction,  multiply  the  quo- 
tient by  the  numerator,  and  place  the  product  over  the  larger 
denominator. 

WRITTEN  EXERCISES. 

Reduce : 

2.  |  to  a  fraction  whose  denom.  is  24.  Ans.  -Jf. 

3.  f  to  a  fraction  whose  denom.  is  42.  Ans.  ff. 

4.  _z_  to  a  fraction  whose  denom.  is  66.  Ans.     ? 

5.  |  to  sixteenths.      Ans.  ff.  |    7.  f  to  fortieths.     Ans.  ? 

6.  f  to  forty-fifths.    Ans.  f  f.  |   8:  T7^  to  sixtieths.    Ans.  ? 

179.  CASE  IV. — To  reduce  a  mixed  number  to  an  im- 
proper fraction. 

1.  Reduce  5|  to  an  improper  fraction. 

EXPLANATION.— AVe  multiply  5  by  3,  add  OPERATION. 

2  to  the  product,  and  place  the  sum  17  over      5  X  3  =  15 
the  denominator.  15  -f~  2=17,  ^. 


Common  Fractions. 


141 


ANALYSIS.— 5f 


Hence,  the 


5  ones  and  2  thirds, 

15  thirds  and  2  thirds  =  17  thirds. 


RULE. — Multiply  the  -whole  number  by  the  denominator, 
add  the  numerator  to  the  product,  and  place  the  sum  over 
the  denominator. 

WRITTEN  EXERCISES. 


Reduce  to  improper  fractions : 


2.  4J. 

3.  7|. 

4.  5f. 

5(~*  O 
•  \  ** 
•         w"fT* 

6.   lOf. 

7         O3 
I  •        t/"?. 


O 

3  ' 
_3_0 

4  ' 

28 


8. 

9.    15f. 
10. 


J-3' 

;s 

>4' 


¥•  j 14- 

6-3    I  15.  45/y. 

50 


Ans. 
Ans. 


11.  ISf. 

12.  17^.     Ans. 

13.  21f.     Ans. 


16. 


17.  60^.     Ans. 

18.  1352jj.  Ans. 

19.  24444.  Ans. 


180.  CASE  V. — To  reduce  an   improper  fraction  to  a 
whole  or  mixed  number. 


l.  Reduce  ^°-  to  a  mixed  number. 

EXPLANATION. — We  divide  the  numerator 
by  the  denominator. 


OPERATION. 

3)10(3i 


ANALYSIS. — ^  -—10  thirds  =  Q  thirds  and  1  Mird 

=  3  and  1  tfiiYd  ==  3}. 

2.  Reduce  2^  to  a  mixed  number. 

ANALYSIS.  —  -2?2-  —  22  fourths 

=  2Qfoiirt)is  and  2fourths=5  and  |. 


OPERATION. 

i22( 

20 


2  -   -  1 


Hence,  the 

RULE.  —  Divide  the  numerator  by  the  denominator,  and  re- 
duce the  fractional  remainder,  if  any,  to  its  lowest  terms. 


142 


Intermediate  Arithmetic. 


WRITTEN  EXERCISES. 

Reduce  to  whole  or  mixed  numbers : 


3.  -V-- 

4.  -y-. 

5.  ^5-. 

ft  ^      - 

•7  48 

I  .  ~ a". 


.    9. 
.  7-. 


Ans. 


.  54. 


8.  -6^.       Ana.  16.  ,   13.  W.  Ans.  131. 

rr  1   O  o 


Ans.  84. 


9. 


10.  ^.       ^4ns. 

11.  i-jp.     ^4m\ 


12. 


78 
T2- 


Ans. 


14.  W.  Ans. 


40 


15.  ^.  ^4m-.       ? 

16.  ^/.  ^4m.    5^. 

17.  -6F2f.  -4na.  12 J. 


181.  CASE  VI.- -To   reduce  a  compound   fraction  to  a 
simple  one. 

1.  Reduce  -f  of  J  to  a  simple  fraction. 

ANALYSIS. — By  Art.  173,  |  of  ^  =  =  T5)  then  f       OPERATION. 
of  i  will  be  2  times  1  fifteenth,  or  T2^,  and  f  of  f    2.  y  i  —  _8_ 


will  be  4  ^m^s  2  fifteenths,  or  T\,  ^4ns.     Hence, 


RULE. — Multiply  the  numerators  together  for  the  numer- 
ator of  the  answer,  and  the  denominators  together  for  the  de- 
nominator. 

EXERCISES. 

2.  Reduce  f  of  f  to  a  simple  fraction.     Ans.  J|  =  -fa. 

3.  Change  f  of  -f$  to  a  simple  fraction.  ylns.  ^. 

4.  Reduce  2  thirds  of  3  tenths  to  a  simple  fraction. 

^4n.s.  1  ^/f/^. 

5.  Change  4  ^/^/is   of  2  sevenths  to  a  simple  fraction. 


6.  What  simple  fraction  is  equal  to  f  of  f  of  f  ? 

Ans.  y\. 

7.  What  simple  fraction  is  equal  to  f  of  -f  of  -}-f. 


8.  Reduce  f  of  7J  of  5  to  a  simple  fraction. 


Common  Fractions.  143 

EXPLANATION.  —  Reduce  1\  to  an  im- 
proper fraction   (-2/)  ;    also   the    whole 

number  5,  by  putting  1  under  it  fora  fX-^-Xf  =  ;Wr  :¥~- 
denominator,  and  proceed  as  before. 

9.  Reduce  f  of  3^  of  \  to  a  simple  fraction.  Ans.  ^. 

10.  Reduce  J  of  -f-  of  9^  to  a  simple  fraction.  Ans.  f. 

11.  What  is  the  value  of  f  of  f  of  l|f?  Ans.  1. 

12.  What  is  the  value  of  J  of  T3T  of  16J?  Ans.  3. 

182.  Cancellation.  Instead  of  multiplying  the  numer- 
ators, then  the  denominators,  and  then  reducing  to 
lowest  terms,  the  same  result  may  be  obtained  by  first 
striking  out  or  cancelling  all  factors  common  to  the 
numerator  and  denominator.  By  this  process  the  work 
is  often  materially  shortened. 

13.  Reduce  f  of  -f  of  f  to  a  simple  fraction. 

EXPLANATION.  —  First,    there   is    a  3  OPERATION. 

in  both  numerator   and   denominator,        q          9          £          9 
one  over  5,  and  the  other  under  2  ;  we  of  _  of  _    —  _ 

cancel  these  by  making  a  mark  across       ft         ft         7          7 
them.     Next,  we  cancel  the  two  5's  in 

the  same  manner.     Cancelling  a  number  is  dividing  by  it  ;  hence, 
1  is  supposed  to  take  the  place  of  the  number  canceled  ;  that  is, 

^121  2 

of  -  of     -  means  -  of     -  of  -.     Hence  the  answer  is  -. 

7  1       1       7  7 


14.  Reduce  f  of  f  of  ^  to  a  simple  fraction. 

EXPLANATION.  -  •  Write    3X3  OPERATION. 

in  the  place  of  9,  3  X  5  in   the  9  a          o  \/  K 

place  of  1-5  ;  then  cancel  as  in  the  -       —   of   -  of  =  1. 

preceding  example.  4 

15.  Reduce  f  of  -J  of  -J-f-  to  a  simple  fraction.    Ans.  f£. 
Reduce  to  a  simple  fraction  : 

16.  |  of  |  of  J  of  6.  Aiis.  3. 


144  Intermediate  Arithmetic. 

17.  f  of  f  of  f  of  &$•  of  f .  Ans.  2. 

18.  f  of  T2T  of  f  of  7J  of  TV  Ans.  £. 

19.  f  of  f  of  J  of  T6r  of  3f  of  8|  of  i  An*.  I. 

183.  CASE  VII. — To  reduce  fractions  to  a  common  de- 
nominator. 

Fractions  have  a  common  denominator  when  their 
denominators  are  the  same :  as  f  and  -f ,  y\-  and  -fa. 

1.  Reduce  |-  and  J  to  a  common  denominator. 

Multiplying  both  terms  of  -|-  by  3,  the  denominator  of  the  other 
fraction,  we  have  i  =  -  |.  Now  multiplying  both  terms  of  \  by  2, 
the  denominator  of  the  other  fraction,  wTe  have  i  =  =  f.  -Hence,  in 
the  place  of  ^  and  |  we  have  f  and  |,  and  these  have  a  common 
denominator. 

Reduce  to  a  common  denominator : 


2.  |  and  -f.    ^4ns.  f-|-  and 


1  I  AJ     t-7  JJ     <J 

3.  f  and  -f.    ylns.  -g8g-  and  -f  J. 

4.  -|-  and  -f.    ^4?is.  f  and  ^ 


5.  i  and  | 


5~- 


6.  |-  and  |.  Ans.  ? 

7.  f  and  ^-. 

8.  Reduce  f,  f,  and  f  to  a  common  denominator. 

EXPLANATION. — We  multiply  both  terms  OPERATION 

of  f  by  7X8,  the  denominators  of  the  4  ^ 
other  fractions,  which  gives  f  -  =  f  f  £.  We 
next  multiply  both  terms  of  f  by  5  X  8,  f  o  X  o  =  :  ^-g-g-. 
the  denominators  of  the  other  fractions,  -f-  5  X  7  —  -^-f-f • 
and  obtain  f-  =  28sV  Similarly  we  get  f  = 

£f  §.     Hence,  instead  of  the  given  fractions  we  have  their  equals; 
Mii,  A0o»  and  4M»  which  have  a  common  denominator. 

AuU/AOl/7  ^i  o  U  / 


Hence,  the 

RULE. — Multiply  both  terms  of  each  fraction  by  the  prod- 
uct of  all  the  denominators  except  its  own. 

NOTE. — Mixed  and  whole  numbers,  if  any,  must  be  reduced  to 
improper  fractions. 


Common  Fractions. 


145 


WRITTEN  EXERCISES. 

Reduce  to  common  denominators  : 


Q    s 

y.  -IT 


LO  common  ueiiuu 

and  f .  Ans.  |f  and  ||. 
1 1.  Ans.  44  and  M. 


10. 

T?0 

an 

idf. 

An 

11. 

f  and  f  .    An 

15. 

3 

T5 

and 

i 

2' 

16. 

3 

2 

35 

and 

f. 

17. 

5 
65 

4 
"55 

and 

I- 

18. 

7 

85 

3 

45 

and 

4 
5' 

19. 

.4  /        O  ' 

20. 

5, 

14,  and  -1 

A  /                          O 

21. 

1, 

-2-5 

and 

2 
5"' 

22. 

2 

"35 

4, 

and 

2J 

1  2.  J  and  T%  . 

13.  -H-  and  |J. 

14.  J^-and  11 


and 


23.  |,  2J,  and  10. 


Ans. 
Ans. 

Ans. 
Ans. 
Ans. 
Ans. 
Ans. 
Ans. 


42 
T6~5 

36 

TTO"5 

75 


40 
5"6"5 

40 
605 

7.1 
905 


II- 

4.5 
6T7- 


TFO' 

JL1Q.     120     J_18. 
1605    1  6~0~5 

J_05_      2.0.      36. 
37  5    305    30- 

5.0.     15.     _6 
105     105    10' 


184,  To  reduce  fractions  to  their  Least  Common  De- 
nominator. 

RULE. — Find  the  L.  C.  M.  of  the  denominators ;  take  it 
for  a  common  denominator,  and  reduce  each  fraction  ac- 
cording to  Case  III. 

24.  Reduce  f,  -f,  and  -f%  to  their  least  common  denom- 
inator. 


EXPLANATION.— The  L.  C.  M.  of  8,  6,  and  12  is       OPERATION. 
24,  which  we  take  for  a  common  denominator. 
Now  by  Case  III  we  say,  8  in  24  3  times,  3X3  = 
9;  hence,  |=  ^V    In  a  similar  manner  we  find 

5  20     nrirl      7     .     -  14 

^--  2T>  ana  TS--  2¥- 

Reduce  to  their  least  common  denominators  : 


3  .9 

8  -  '  24 

A  2. 0. 

6  '  "  24 

JL  14. 

12  '  "24 


25, 
26. 
27. 
28. 


i          o          o  T\  r\       O 

"35    4"5    ana    6"' 

2.     4.     5. 

35    95    65 

1      3        5          7 
45    85     165    32' 
_3       _5_     _7          < 
175     125    13"5 

J.  N.— 10. 


4?1S. 

Ans. 
Ans. 
Ans. 


8 
125 

12 

185 

8 


9 

12") 

8 
185 

12 
•"" 


18 

"6"o"5 


2S 

" 


10 
12' 

15 

185 

10 
3T5 

28 

"605 


7 
18' 


27 

fir- 


146 


Intermediate  Arithmetic. 


ADDITION  OF  FRACTIONS. 


INDUCTIVE  EXERCISES. 
185.  PARALLELISMS. 


WHOLE  NUMBERS 

1.  Add: 

4  threes  and  7  threes. 

Ans.  11  threes. 

2.  Add: 

8  threes  and  9  fours. 
These  are  unlike,  and  must 
be  reduced  to  like  units. 

8  threes  =  2  twelves. 

9  fours  =  3  twelves. 

Now  adding  the  like  num- 


bers, we  get 

5  twelves. 


FRACTIONS. 

1,  Add: 

4  thirds  and  7  thirds. 

Ans.  11  thirds. 

2.  Add: 

8  thirds  and  9  fourths. 
These  are  unlike,  and  must 
be  reduced  to  like  units. 

8  thirds  =  32  twelfths. 

9  fourths  =  27  twelfths. 
Now  adding  the  like  num- 


bers, we  get 

59  twelfths. 

PRINCIPLE. —  To  add  two  numbers,  whether  whole  or  frac- 
tional, they  must  be  reduced,  if  not  already  so,  to  like  units. 


186.  CASE  I. — When  the  denominators,  or  fractional 
units,  are  alike. 


1.  Add  together  f,  -f-,  and  ±. 

EXPLANATION. — Since  the  numbe 
added  have  the  same  unit,  viz  :  1  seventh,  we 


OPERATION. 

f  =  3  sevenths. 
EXPLANATION. — Since  the  numbers  to  be    2  __  o  seventh'* 

add  as  in  whole  numbers,  and  obtain  6  sev-    Z_m 

enths,  or  f.    Hence,  6  sevenths^. 


RULE. — Add  the  numerators  and  place  the  sum  over  the 
common  denominator. 


Addition  of  Fractions. 


147 


What  is  the  sum  of: 

2.  f ,  f ,  |  ?          ylns.  f  = 

3. 

4. 

5. 


346?  Am<i 

T)  T)  T  jaUiio* 

.4      5.     1     8.  9 

9)    9)    9)    9   * 

_7_     _2_     _5_       4 
11)    11)    11)    11 


=  1i- 

6. 

&, 

TV, 

TV, 

4 
TU 

? 

.4ns. 

If. 

7. 

A, 

ii, 

TV, 

A 

? 

,4ns. 

2. 

8. 

3     3.     5 

¥)  ¥>  ¥ 

? 

Ans. 

l-7- 

•  ATT' 

9. 

$t> 

S-V 

-,  H 

? 

Ans. 

OPERATION. 
9 


187.  CASE  II. — When  the  denominators,  or  fractional 
units,  are  unlike. 

l.  Add  together  f  and  f. 

EXPLANATION. — Since  the  numbers  to  be 
added,  viz  :  3  fourths  and  2  thirds,  are  unlike, 
they  cannot  be  added  in  their  present  form. 
Reducing  them  to  a  common  denominator, 
or  unit,  by  Art.  183,  we  obtain  ^-2-  and  T\, 
the  sum  of  which,  by  Case  I,  is  j|  =  lj\.  Hence, 

RULE. —  Reduce  the  fractions  to  a  common  denominator, 
and  proceed  as  in  Case  L 

NOTE.  —  In  addition  and  subtraction,  the  fractions  should  be 
written  under  each  other  after  the  manner  of  whole  numbers. 


3  

f : 

2  8 

¥-  -jnr 

1  7 


What  is  the  sum  of 


and 
and 
and 

7.  -fi  and 

8.  I-  and 


4. 
5. 
6. 


3 

6" 
5 

8" 


f? 


f? 


A? 


Ans. 
Aiis. 
Ans. 

Ans. 
Ans. 


f  f  and 

1? 

^4ns 

3.1  _ 

=  1 

i.i 

f  f  and 

2 
3 

? 

Ans. 

113. 
24* 

A- 

9. 

2 
3) 

3 

T> 

and 

6. 
~5 

? 

Ans. 

2 

f-J- 

45 
42" 
93 

T2- 

*f 

10. 
11. 
12. 

1, 

5 

6") 

5 

6) 

and 
and 
and 

1 
"8 
8 
•&" 

? 

• 

? 
? 

Ans. 
Ans. 

037 

^8*' 

2  95 

O  3  1  3 
Z5"OT' 

11 

1 

3. 

1! 

8 

"Q"l 

and 

TT 

.? 

• 

Ans. 

24 

M- 

14.  Find  the  sum  of  7|  and  9J. 

EXPLANATION. —  When  there  are  mixed 
numbers,  we  add  the  fractions  first,  and  then 
add  their  sum  to  the  sum  of  the  whole  num- 
bers. Adding  T92  and  T82,  we  get  H  =  lT6s ; 
put  down  the  T\  and  carry  1  to  be  added  to 
the  sum  of  the  whole  numbers,  we  get  17y\. 


OPERATION. 

8 


72  _  7 

ft~    -  *J 
Q3  -  Q  9 


TT- 


148 


Intermediate  Arithmetic. 


15.  Add  together  12J  and  15J.  Ans.     27f. 

16.  Add  together  23f,  18f,  and  32|.  ^?is.  74ji. 
17   A  man  paid  $13J  for  a  pair  of  pants,  $17f  for  a 

coat,  and  $5f  for  a  vest.     What  did  he  pay  for  all? 

18.  One   boy   weighs    64f  pounds,   another   boy   56f, 
pounds,  and  the  third  boy  49^  pounds.      What  is  the 
total  weight  of  the  three  boys?         -4ns.  170JJ  pounds. 

19.  A  man   planted  120f   acres  in  corn,  75  J  acres  in 
cotton,    32^   acres    in    wheat,    and   15fV   acres    in   oats. 
How  many  acres  did  he  have  in  cultivation? 

Ans.  243      acres. 


SUBTRACTION  OP  FRACTIONS. 


INDUCTIVE  EXERCISES. 


188.  PARALLELISMS. 


WHOLE  NUMBERS. 

1.  From 

4  fives  take  2  fives. 

Ans.  2  fives. 

2.  From 

15  fours  take  8  fives. 

These  are  unlike,  and  must 
be  reduced  to  like  units. 

15  fours  =  3  twenties. 
8  fives  =  2  twenties. 

Now  subtracting  like  num- 
bers, we  get 

1  twenty. 


FRACTIONS. 

1.  From 

4  fifths  take  2  fifths. 

Ans.  2  fifths. 

2.  From 

15  fourths  take  8  fifths. 

These  are  unlike,  and  must 
be  reduced  to  like  units. 

15  fourths  -  -  75  twentieths, 
8  fifths  =  32  twentieths. 

Now  subtracting  like  num- 
bers, we  get 

43  twentieths. 


Subtraction  of  Fractions.  149 

PRINCIPLE. --To  subtract  one  number  from  another,  whether 
whole  or  fractional,  they  must  be  reduced,  if  not  already  so,  to 
like  units. 

189.  CASE  I. — When  the  denominators  or  fractional 
units  are  alike. 

1.  Subtract  T%  from  ^. 

ANALYSIS. — Since  the  numbers  to  be  subtracted,  viz :  3  tenths 
and  9  tenths,  have  the  same  unit,  1  tenth,  we  subtract  as  in  whole 
numbers,  and  obtain  6  tenths,  or  T%  =  f ,  Ans. 

Hence,  the 

* 

RULE. —  Take  the  less  numerator  from  the  greater,  and 
place  the  difference  over  the  common  denominator. 

2.  What  is  the  difference  between  f  and  f  ? 

3.  What  is  the  difference  between  ^  and 

How  much  more  is : 


7.  --    than      -?         Ans.  ? 


8.  1^  than  T\?         Ans.  ? 

9.  A!  than  4-1- ?         Ans.  ? 


4.  T7-g  than  y^?     Ans. 

5.  if  than  T72-?      Ans.  f. 

6.  i|  than  A  ?      Ans.  \. 

1    '.)  LI)  O  —     -i  —     » 

190.  CASE  II. — When  the  denominators  or  fractional 
units  are  unlike. 

l.  From  f  take  %. 

EXPLANATION. — Since  3  fifths  and  1  Aa?/are  un-       OPERATION. 
like,  they  can  not  be  subtracted  in  their  present       3 
form.    Reducing  them  to  a  common  denominator,        i 
by  Art.  183,  we  obtain  6  tenths  and  5  tenths,  the 
difference  between  which  is  1  tenth,  or  y1^.  yV  Ans. 

Hence,  the 

RULE. — Reduce  the  fractions  to  a  common   denominator, 
and  proceed  as  in  Case  I. 


150 


Intermediate  Arithmetic. 


Find  the  difference  between  : 

2.  f  and  f. 

3.  f   and  }. 

4.  -J  and  J. 

5.  f   and  f. 

6.  T9T  and  TV 

7.  y9^  and    f 

8.  |    and  i2T. 

16.  From  8f  take  5|. 

EXPLANATION. —We  first  reduce  the  fractions 
to  a  common  denominator,  then  take  their  differ- 
ence, and  unite  it  to  the  difference  of  the  whole 
numbers.  Thus,  -^  from  T9j  leaves  y1^ ;  5  from  8 
leaves  3  ;  now  uniting  the  3  and  y1^,  we  get  3^2. 


5  ^ 

9. 

2" 

and 

24' 

Ans. 

II 

3 

28' 

10. 

3. 

and 

T3~' 

Ans. 

H- 

1 

11. 

5 
TT 

and 

2. 

Ans. 

23 
99 

T8-- 

12. 

10 
13 

and 

3 

8' 

Ans. 

? 

yW 

13. 

1 
4 

and 

3 

Ans. 

? 

• 

14. 

2 

and 

10 
43' 

Ans. 

? 

? 

15. 

H 

and 

A- 

Ans. 

? 

OPERATION. 

8!  =  8A 

8 


From  : 

17.  12f  take    7|. 
18. 


take  18J.  Ans. 


19.  351  take  22f. 

20.  27    take  23  .  Ans.    4. 


21.    451  take  71.     Ans.  38M. 


24' 


22.  1641  take  73f          Ans.t 

23.  195ttakel26TV      Ans.t 

24.  200^  take  85 A-.        Ans.  ? 


25.  A  rope  was  48J  feet  long,  but   17|  feet  were  cut 
off;  how  long  was  the  rope  then? 

26.  From  8  take  I. 


EXPLANATION. — 8  is  equal  to  7  and  one,  or,  re- 
ducing one  to  ninths,  7  and  f .  Hence,  we  write  8 
under  the  form  of  7f,  and  subtract  f  as  in  the 
preceding  examples. 

How  much  more  is  : 


OPERATION. 

8  =  7* 

5  5 

9"  9 


71. 


27.  12  than  J?   Ans.  llf 


28.  13  than  -f^  Ans.  12T6T. 

29.  14  than  f  ?   Ans.  13|. 

30.  43  than  ^  Ans.  42T%. 


31.  64thanT3T? 

32.  75  than  ^? 

33.  84  than  Ty? 

34.  125  than -i? 


Ans.  ? 
Ans.  ? 
-4ns.  ? 
Ans.  ? 


Subtraction  of  Fractions.  151 

35.  From  7J  take  5f. 

EXPLANATION.— We  reduce  the  fractions  to  a  OPERATION. 

common  denominator,  and  as  we  can  not  subtract  7-g-  =  7^2 

A  from  T\,  we  take  one,  or  }•?„  from  the  7,  and  add  ^3  -  -  c;  9 

i_i  ±  &/  x—  *-^  4.    12 

it  to  'T42>  making  ||.     Then  we  say,  T\  from  || 

leaves  T72,  and  5  from  6  (7  less  one)  leaves  1.  lyV 


What  is  the  value  of: 

36.  8£  —  3f? 

37.  16f  —  12f  ? 

38.  43J  — 18f?  ,4ns.  24^. 


39.  68|-  -49J?    And. 

40.  lOO-  - 


41.  146f  — 86}f: 

42.  A  man   had   $5J  and   paid   for  a   knife  $f ;   how 
much  did  he  have  left?  Ans.  $4^. 

43.  One    melon    weighs    25^    pounds,   and    another 
weighs    17T\   pounds  ;    how   much   heavier  is  one  than 
the  other?  Ans.  7f££  pounds. 

44.  Frank    and    John    went    fishing ;    Frank    walked 
IS^V  miles  and  John  2g\  miles ;  how  much  further  did 
Frank  walk  than  John?  Ans.  15f|f  miles. 

45.  A  farmer  sold    If  acres   from  a  field   containing 
37f  acres  ;  how  many  acres  had  he  left  in  the  field  ? 

Ans.  35|-f  acres. 

46.  One  bale  of  cotton  weighs  463^   pounds,  and  an- 
other   bale    weighs    17f    pounds    less ;    what    does   the 
lighter  bale  weigh  ?  Ans.  ? 

191.  In  the  following  examples  the  whole  and  one  or 
more  parts  are  given,  and  the  c  part  required.  See 
Art.  89. 

47.  The  whole  is  385J  days,  and  one  part  67^  days; 
what  is  the  other  or  c  part?  Ans.  317j^  days. 

48.  A  man  had  $1673f  and  spent  $356f ;   how  many 
dollars  had  he  left?  Ans.  $1316|J. 


152 


Intermediate  Arithmetic. 


49.  A  man  had  S540J-  and  spent  $271f  for  sheep,  and 
8180J  for   hogs;   how   much   money  did   he   have  left? 

Ans.  $88TV. 

50.  A   flag-pole,    standing   in    the   water,    is    100   feet 
long;  72 J  feet  of  its  length   are   above   the  water, 'and 
12f  feet  are  in  the  mud  below  the  water ;  how  deep  is 
the  water?  Ans.  14fJ  ft. 

51.  Two  boys,  Charles  and  Henry,  are  400  feet  apart; 
if  Charles   goes   towards   Henry  167f   feet,  and   Henry 
goes  towards  Charles  207f  feet,  how  far  apart  will  they 
then  be?  Ans.  24^  ft- 


MULTIPLICATION  OP  FRACTIONS. 

INDUCTIVE  EXERCISES. 
192.  PARALLELISMS. 


WHOLE  NUMBERS. 

1.  Multiply  : 

5  threes  by  4  threes. 

Since  5  X  4  —  20,  and 
three  X  three  =  nine, 
5  threes  X  4  threes  — 

20  nines. 

2.  Multiply: 

7  threes  by  5  fours. 

Since  7  X  5  —  35,  and 
three  X  four  =  twelve, 

7  threes  X  5  fours  = 
35  twelves. 


FRACTIONS. 

1.  Multiply: 

5  thirds  by  4  thirds. 

Since  5  X  4  —  20,  and 
1  third  X  1  third  =  1  ninth, 
5  thirds  X  4  thirds  = 

20  ninths. 

2.  Multiply  : 

7  thirds  by  5  fourths. 

Since  7  X  5  =  35,  and 

1  third  X  1  fourth  = 
1  twelfth, 

7  thirds  X  5  fourths  = 
35  twelfths. 


Multiplication  of  Fractions.  153 

PRINCIPLE. —  To  multiply  two  abstract  numbers,  whether 
whole  or  fractional,  we  may  multiply  the  numbers  regarded 
as  units  together  for  a  new  unit,  and  the  numerators,  or 
numeral  factors,  together  for  a  new  numerator. 

193.  CASE  I. --To  multiply  a  fraction  by  a  whole 
number. 

1.  Multiply  f  by  3. 

EXPLANATION. -- The   multiplicand   is   5        1st.  OPERATION. 
sixths  and  the  multiplier  3.     3  times  5  sixths    5^0     .  i  s  _  -  91 

i  -     ..  tf  X  O  -      -g-      -  ZT. 

are  lo  sixths,  or  -^    =  2$. 

Instead  of  multiplying  the  factor  5  by  3, 
we  may  multiply  the  unit  factor  1  sixth  by 

,50     0    ,.  i      •    *v    •      i    i.    i*  2d.    OPERATION. 

3.    By  Art.  173,  3  times  1  sixth  is  1  half. 

Hence,  5  sixths  X  3  =  5  halves,  or  f  =  =  2$.       f  X  3  =  f  =  2£. 

Hence,  the 

RULE. — Multiply  the  numerator  by  the  whole  number,  or 
divide  the  denominator  by  the  ivhole  number  when  it  can  be 
done  without  a  remainder. 

EXERCISES. 

Multiply : 


. 


2.  |   by  3.         Ans.     1| 

3.  |-  by  4.         Ans. 


8-  H  by    8. 

9.  ||  by    7.          Ans.  2J. 


10.  ff  by  11. 

11.  f£  by  12.  Ans.     ? 

12.  ^  by  10. 

13.  if  by 


4.  ^-  by  6.  ,4ns. 

5.  f  by  7.  Ans.  4. 

6.  -5^-  by  8.  Ans.  1%. 

7.  JJ-  by  5.  -4ns.  ? 

14.  If  1  gallon  of  syrup  cost  f  of  a  dollar,  what  will 
6  gallons  cost?  Ans.  $3f. 

15.  If   1  bushel  of  potatoes  cost  f  of  a  dollar,  what 
will  8  bushels  cost?  Ans.    ? 

16.  If  1  basket  holds  f  of  a  bushel  of  corn,  how  many 
bushels  will  8  baskets  hold?  Ans.  6f  bushels. 


154  Intermediate  Arithmetic. 

17.  If  1  bucket  holds  -f-   of  a  gallon    of  water,  how 
many  gallons  will  5  buckets  hold  ?  Ans.     ? 

18.  Multiply  7}  by  5. 

We  may  reduce  7|  to  an  improper  frac-  OPERATION. 

tion,  and  multiply  as  in  the  preceding  exer-  «3 

cises.     It  is  generally  better,  however,  to  ^ 

multiply  thus  :  5  X  f  =  =  ¥  =  =  3!-    5  X  7  ==  35.  _A 

Now,  adding  3|  to  35,  we  have  38f. 

35 


NOTE.— Since  8]-  X  5  =  5  X  8},  it  is  imma- 


terial which  we  regard  as  the  multiplier.  38f 

What  is  the  value  of: 


19.     6-|  X    7?  Ans.     46|. 


20.  7|X    3?    Ans.     22^ 

21.  12f  X    9?   Ans.  115^ 

22.  10}  X    8?       Ans.      i 


23.     7fX20?       Ans. 


24.     9X    18|?       Ans.  168. 


25.  12  X    13f?       Ans.  166. 

26.  20  X    llf?  Ans-   ? 

27.  48x30yV?  Ans.   ? 


28.  72  X  1234?  Ans.   ? 


29.  What  will  32  gallons  of  brandy  cost  at  $lj-  per 
gallon?  Ans.  $36. 

30.  What  will   IS-j^  barrels  of  apples   cost  at  $3   per 
barrel?  *  Ans.  $45^. 

31.  What  will  556  pounds  of  cotton  amount  to  at  8f 
cents  a  pound  ?  Ans.  4865  cents. 

What  will  be  the  cost  of: 

32.  654  pounds  of  cotton  at  7|  c.  a  pound  ?  Ans.  5014  c. 

33.  255  pounds  of  sugar  at  9f  c.  a  pound?  Ans.  2448  c. 

34.  876  yards  of  prints  at  5}  c.  a  yard  ?        Ans.  5037  c. 

35.  1260  bushels  of  corn  at  64|  c.  a  bushel  ?  Ans.  81480  c. 

36.  570  yards  of  silk  at  $1|  a  yard?  Ans.  $798. 

194.  CASE  II. — To  multiply  a  fraction  by  a  fraction. 
1.  Multiply  f  by  J. 


Multiplication  of  Fractions.  155 

EXPLANATION. — Multiplication  means  tak-  OPERATION. 

ing  one  number  as  many  times  as  there  are    5sx_l--5X2-.io 

•       •  "7  /^  U  —  TTxTT  —  ^T* 

units  in  another.     In  f  there  are  only  f  of  a 

unit.     Hence,  we  are  required  to  take  f  f  of  1  time.     Now,  1  time 

f  is  f ,  hence  f  of  1  time  f  is  f  of  f  =  |-J. 

Hence,  the 

RULE. — Multiply  the  numerators  together  for  a  numera- 
tor, and  the  denominators  together  for  a  denominator. 

Or,  Regard  X  as  meaning  of,  and  proceed  as  in  the  re- 
duction of  compound  fractions  to  simple  ones. 

NOTE. — Mixed  numbers  must  be  reduced  to  improper  fractions. 
Multiply : 


2.  |  by  Ans. 

3.  i  by    f        Ans. 


4.     f  by    f.        Ans.     J, 


5- 


- 


7J  by  8f 


5.      ^n«.      42. 


. 


9.  6f  by  31.      Ans. 

10.  7f  by  2f.      Ans.  20^. 

11.  4^  by  2J. 


6.  4J  by    |. 

What  will  be  the  cost  of: 

12.  12  yards  of  cloth  at  8  c.  a  yard?  Ans.  96  c. 

13.  4  barrels  of  cider  at  $3£  a  barrel?  Ans.  $13. 

14.  25|  pounds  of  coffee  at  12  c.  a  pound  t  Ans.  308  c. 

15.  If  barrels  of  flour  at  S5f  a  barrel?       Ans.  $10ff. 

16.  9f  bushels  of  corn  at  62J  c.  a  bushel?  Ans.  570^  c. 

17.  240y\  acres  of  land  at  $25f  an  acre?  Ans.  $6156ff. 

What  is  the  cost  of: 

18.  16  pounds  of  cheese  at  8|  c.  a  pound?  Ans.  136  c. 

19.  15|  yards  of  cambric  at  15  c.  per  yard?  Ans.  235  c. 

20.  11£  cords  of  wood  at  $3J  per  cord?         Ans.  $38|. 

21.  15 \  yards  of  broadcloth  at  $3|^-  a  yard?  Ans.$57^%. 

22.  15f  yards  of  ribbon  at  40  c.  per  yard?    Ans.  630 c. 

23.  8 J  yards  of  silk  at  $^  per  yard?  Ans.  84f. 

24.  348  pounds  of  cotton  at  7|  c.  per  pound  ? 


156 


Intermediate  Arithmetic. 


DIVISION  OF  FRACTIONS. 


INDUCTIVE  EXERCISES. 


195.  PARALLELISMS. 


WHOLE  NUMBERS. 

1.  Divide 

15  fours  by  3  Jours. 

Ans.  15-r-3=5. 

2.  Divide 

15  fours  by  8  Jives. 

Reducing   to  like   units : 

15  fours  =  3  twenties, 
8  fives  =  2  twenties. 

Now, 

3  twenties  -r-  2  twenties 
=  3  -f-  2  —  U. 


5. 


FRACTIONS. 

1.  Divide 

15  fourths  by  3  fourths. 
Ans.  15  -j-  3 

2.  Divide 

15  fourths  by  8  fifths. 

Reducing   to   like  units : 

15  fourths  =•-  75  twentieths, 
8  fifths  =  32  twentieths, 

Now, 

75  twentieths  -?-  32  twentieths 
=  75-^32  =  211 


PRINCIPLE. —  To  find  how  often  one  number  is  contained 
in  another,  whether  whole  or  fractional,  they  must  be  reduced, 
if  not  already  so,  to  like  units. 

196.  CASE  I. — To  divide  a  fraction  by  a  whole  number. 
1.  Divide  *£•  by  5. 

1st  OPERATION. 


EXPLANATION.— The  dividend  is 
20  thirds,  the  divisor  5.     Now, 
20  thirds  -f-  5  =  4  thirds  ==  f  =  =  1$. 
Instead  of  dividing  the  factor  20 
by  5,  we  may  divide  the   unit  1       20  _^  F;  _  _  20.  —  4  —  11 

~3™     '          T  5  U  ~    ~      3  * 

third  by  5.     By  Art.  173,  1  third 

-*-  5  =  =  1  fifteenth.   Hence,  20  thirds  -4-  5  ==  20  fifteenths  ==  f  §  =  f  =  1$. 


o    •    P; 4  —  11 

Q —    O     -K-    J.-5-. 


2d    OPERATION. 


Division  of  Fractions.  157 

RULE. — Divide  the  numerator  or  multiply  the  denominator 
by  the  whole  number. 

EXERCISES. 

Divide : 

2.  f  by  2.  Ans.  J.  8-  if  by  6. 

3.  f  by  2.  Ans. 

4.  f  by  6.  Ans. 

5.  12  fifths  by  4.    ^4ws.  f. 

6.  15  halves  by  5.  ^4?is.  1|-. 

7.  ft  by  9.  Ans. 

1  o        «/ 


3 
8' 

4 
2T- 


/;     4        1  /^  j  /~v  rt 

9.  ^  by  8.  ^Lns.  2|. 

10.     f  by  9.  Ans.  -fa. 

11          1  0      V»T7     19  /J  1}  <3         ? 

1>  TS"  DJ  1Z-          -fins,   i 

12.  ff  by  25.         Ans.  ? 

13.  7  ninths  by  4.  J.TIS.  -j7g-. 

14.  If  4  tops  cost  $f,  what  will  1  top  cost?    Ans.  %%. 

15.  If  3  melons  cost  $f,  what  will  1  melon  cost? 

Ans.  $-f-. 

16.  If  5  spellers  cost  S^-J,  what  will  1  speller  cost? 

Ans.  $T2T. 
CASE  II.- -To  divide  1  by  a  fraction. 

INDUCTIVE  EXERCISES. 

197.  Divide  a  bar  of  soap  into  three  equal  parts,  thus : 

How  often  are  2  parts  con- ^^^ 

tained  in  3  parts  f       Ans.  f . 
What   stands   for   2  parts? 


What  stands  for  3  parts?  Ans.  1. 

How  often,  then,  is  |  contained  in  1  ?  Ans.  f  . 

How  often  is  f  contained  in  1  ?  Ans.  |. 

Why  ?  Ans.  Because  f  -  =  3  fourths,  and  1  =  4  fourths, 
and  4  fourths  -j-  3  fourths  =  ^. 

How  often  is  f  contained  in  1  ?  Ans.  %  times,  because 
5  sevenths  is  contained  in  7  sevenths  -J 


198.  The  Reciprocal  of  a  Fraction  is  the  result  of  in- 
terchanging the  places  of  its  terms.  The  reciprocal  of 
|  is  f  ;  of  f,  |;  of  f  £;  of  4,  -J-  ;  of  2£  or  J,  f  ;  etc. 


158  Intermediate  Arithmetic. 

Inverting  a  fraction  is  taking  its  reciprocal. 

• 

199.  From  the  preceding  articles  we  derive  the 

RULE.  —  To  find  how  often  a  fraction  is  contained  in  one, 
or  1,  we  take  its  reciprocal,  or  invert  it. 

EXERCISES. 

How  often  is  : 

1.  f  contained  in  1?  Ans.  f  times. 

2.  ^  contained  in  1  ?  Ans.  {%  times. 

3.  f  contained  in  1  ?  Ans.  f  times. 

4.  f  contained  in  1  ?  Ans.  2  times. 

5.  Is  |  equal  to  i?     Why? 

Ans.  Because  each  is  contained  in  1  2  times. 

6.  Is  T\  equal  to  ^?     Why? 

7.  Is  -fa  equal  to  ^?     Why? 

8.  Is  3^  equal  to  J|?     Why? 

How  many  : 

9.  |  will  it  take  to  make  1  ?  Ans.  4. 

10.  T\  will  it  take  to  make  1?  Ans.  5. 

11.  -iV  will  ^  take  to  make  1?  Ans.  4J. 


200.  CASE  III.  —  To  divide  a  whole  number  or  a  frac- 
tion by  a  fraction. 

1.  Divide  5  by  f. 

EXPLANATION.  —  By  Case  II.  f  is  contained  OPERATION. 

in  1  |  times  ;  hence,  in  5  it  is  contained  5     JL  v  &  —  1  5  —  71 

l    /^   2  ~         2         '   '  '2 

times  f  or  *£-  —  7|. 

2.  Divide  f  by  f. 

EXPLANATION.—  By  Case  II,  f  is  contained  OPERATION. 

in  1  1  times  ;  hence,  in  f  it  is  contained  f     8vl--H-- 
\         01  ¥  X  5---ZTF- 

times     or=l. 


Division  of  Fractions. 


159 


Hence,  the 

RULE. — Invert  the  divisor  and  proceed  as  in  multiplica- 
tion of  fractions. 

NOTE. — This  rule  is  also  applicable  to  Case  1. 

EXERCISES. 


Divide  : 

3. 

18 

by 

2 

Ans. 

27. 

10. 

H 

by 

•2"' 

Ans. 

3. 

4. 

20 

by 

i- 

Ans. 

80. 

11. 

3i 

by 

2^ 

Ans. 

5. 

5. 

6 

by 

C   • 

Ans. 

10. 

12. 

5J 

by 

3 

Ans. 

7. 

6. 

15 

by 

5 
8"' 

Ans. 

24. 

13. 

o  i 

by 

I- 

Ans. 

3. 

7. 

I 

by 

f- 

Ans. 

32 

2T- 

14. 

71 
i  2 

bv 

*t 

4- 

Ans. 

? 

8. 

1 

by 

3 

Ans. 

5 
6* 

15. 

5i 

by 

3|. 

Ans. 

l\ 

9. 

3 

T 

by 

2 

Ans. 

9 

16. 

6J 

by 

If- 

Ans. 

3i 

17.  Divide  13J  by  7£. 

18.  Divide  24|  by  2f. 

19.  Divide  15f  by  3f. 

20.  How  many  f  make  7J? 

21.  How  many  If  make  14|? 

22.  How  many  $f  make  SIOJ? 

23.  How  many  Sf  make  $f  ? 

What  will  1  yard  of  cloth  cost : 

24.  If  4  yards  cost  12  cents? 

25.  If  4  yards  cost  12|-  cents? 

26.  If  4|-  yards  cost  12  cents? 

27.  If  4|  yards  cost  12^-  cents? 

28.  If  3J  yards  cost  13J  cents? 

29.  If  5J  yards  cost  31|-  cento? 

30.  If  3J-  yards  cost  $|  ? 

31.  If  51  yards  cost  $f  ? 

32.  If  3|  yards  cost  $2f  ? 

33.  If  5£  yards  cost 


10. 

Ans.  12. 


Ans.  2c 


4c. 
Ans.    ? 
-4ns. 
-4ns.    ? 
Ans.    ? 


160  Intermediate  Arithmetic. 


WRITTEN  EXERCISES. 

201.  An  important  class  of  problems  invoking  Multiplica- 
tion and  Division  of  Fractions.     (See  Art.  142.)  * 

1.  If  If  yards  of  cloth  cost  $lf,  what  will  2i  yards 
cost? 

EXPLANATION,  —  Since  If  yards  cost  OPERATION. 

$1|,   we  divide  If  by  If  to  get  the  j£  _._  j  2  -  -  n. 

cost  of  1  yard,  which  gives  $ft.     Now  ^  v  91       ill       o  ju 

since  1  yard  costs  $|f,  we  multiply  20  A  ^5  -  -  100  -     4ioTT- 
f  ^  by  2i  to  get  the  cost  of  2i  yards, 
and  obtain                            v>2jYo>  Ans. 

2.  If  2J-  yards  of  cloth  cost  $3f,  what  will  If  yards 
cost?  Ans.  $2-J. 

3.  If  f  yard  of  cloth  cost  21  cents,  what  will  2-f  yards 
cost?  Ans.  60  cents. 

4.  If  2£  yards  of  cloth  cost  $lf,  what  will  2J  yards 
cost?  Ans.  $1^-. 

5.  If  3^  yards  of  cloth  cost  $4f,  what  will  2J  yards 
cost?  Ans.  S3. 

6.  If  f  yard  of  cloth  cost  3J  cents,   what  will   13£ 
yards  cost?  Ans.  ? 

How  much  will  : 

7.  12  yards  of  cloth  cost  if  3|-  yards  cost  21  cents? 

Ans.  72  cents. 

8.  15^-  bushels  of  corn  cost  if  3J  bushels  cost  $lf? 

Ans.  $6.51. 

9.  7^  gallons  of  syrup  cost  if  2J  gallons  cost 

Ans. 
10.  13^-  pounds  of  beef  cost,  if  ^  pounds  cost  H  c.  ? 


=:'-  Before  the  pupils  begin  these  exercises,  let  them  solve  six  or  eight  of 
the  examples  under  Art.  142. 


Division  of  Fractions.  161 

11.  How  much  will  25^  acres  of  land  cost,  if  2|-  acres 
cost  $26  ?  Ans.  $304. 

12.  How  much   will    10}  barrels  of  flour   cost,   if  l^ 
barrels  cost  $8f  ?  .  Ans.  $86. 

13.  A  farmer  bought  6J  pounds  of  nails  for  25  c.,  and 
desires  to  get  8J  pounds  more  at  the  same  rate;   how 
much  will  they  cost?  Ans.  32  c. 

14.  A  farmer's  price  for  250^-  acres  of  land  is  $2505 ; 
what  is  his  price  for  175}  acres?  Ans.  $1757 J-. 


202.  PARALLEL  PROBLEMS. 


1.™  Reduce  to  lowest  terms  T6¥,  -J-f, 
2.     Reduce  to  lowest  terms  Iff,  TVoV  Ans.  f,  -f. 

3.m  Change  f  to  twelfths  ;  }  to  twentieths. 
4.     Change  -JJ  to  one  hundred  sixty  -eighths.    Ans.  -}-j-f. 
5.™  What  is  the  number  whose  seventh  is  12? 
6.     What  is   the  number   whose   thirty-sixth    is  95? 

Ans.  3420. 

7.™  Six-sevenths  of   a  number   is    12  ;    what   is  the 
number  ? 

8.     ff  of  a  number  is  138;  what   is   the   number? 

Ans.  282. 

9.™  What  is  the  weight  of  a  beef,  if  }  of  it  weighs 
600  pounds  ? 

10.     What  is  the  weight  of  a  beef,  if  ff  of  it  weighs 
629  pounds?  Ans.  697  pounds. 

11.™  If  f  of  an  acre  of  land  cost  $12,  what  will  one 
acre  cost? 

12.     If  |f  of  a  lot  is  valued    at   $3225,   what  is  the 
price  of  the  whole  lot  ?  Ans.  $3750. 

13.™  If  3  fourths  of  a  pound  of  sugar  cost  9  cents,  what 
will  1  fourth  of  a  pound  cost? 

N.  I.—  11. 


162  Intermediate  Arithmetic. 

14,  If  f-f  of  a  load  of  cotton  cost  $175,  what  would 
of  the  load  cost?  Ans.  $7. 

15.  Reduce  ^8-  to  a  mixed  number.  Ans.  31f. 
6.m  What  is  the  sum  of  -J-  and  -^?    ^  and  ^? 


17.     What  is  the  sum  of  -f  and  ff  ? 

What  is  the  number  whose  c  parts  are  : 

18  m  5  and  |?    7£  and  J?    9f  and  3^?    6J  and  5|? 

19.  18fand36fV?    481f  and  196f  ?    Ans.  55J,  678i|. 

20.  A  merchant  has  two  pieces  of  prints,   one  con- 
tains 23f  yards  and  the   other   48f  yards  ;   how  many 
yards  in  both  pieces  ?  Ans.  72^J-  yards. 

21.  m  What  is  the  difference  between  ^  and  ^?  7  and  f  ? 

22.  What  is  the  difference  between  |f  and  -^J  ?  Ans.  f-^. 

If  one  of  the  parts  of: 

23.  m  8  is  1J,  what  is  the  c  part? 

24.  12  is  7|,  what  is  the  c  part?  Ans.  4|. 
25.™  $8J  is  $5,  what  is  the  c  part? 

26.  A  man  had  $125f  and  spent  $83  J  ;   how   much 
had  he  left  ?  Ans.  $42^-. 

27.  From  134J  gallons  of  water  there  were  drawn  off 
117f  gallons;  how  many  gallons  were  left?     Ans.  16^. 

28.™  If  two  of  the  parts  of  8  are  2  and  3J,  what  is 
the  c  part? 

29.  If  two  of  the    parts  of  165f   are   17^  and   88f, 
what  is  the  c  part?  Ans.  59^-. 

30.  Three  boys   together  weigh   210J  pounds.      The 
first  boy  weighs  71|  pounds,  and  the  second  69|  pounds  ; 
what  does  the  third  boy  weigh  ?          Ans.  69j^  pounds. 

31.*^  What  will  be  the  cost  of  9  apples  at  2  c.  apiece? 
Of  12  peaches  at  f  c.  apiece  ?  Of  8  gallons  of  syrup  at 
$|  a  gallon  ?  Of  9  bushels  of  corn  at  If  a  bushel  ? 

32.  What  will  568  pounds  of  cotton  cost  at  9|  c.  per 
pound?  Ans.  5538  c. 


Division  of  Fractions.  163 

33.  What  will  18J  pounds  of  butter  cost  at  18f  c.  per 
pound?  Ans.  351^  c. 

34.  What  is  the  number  whose  c  factors  are  5J  and 
5|?  Ans.  27. 

35.™  What  will  be  the  cost  of  1  yard  of  cloth  if  4 
yards  cost  $|f  ?  If  4  yards  cost  $f  ? 

36.  What  will  be  the  cost  of  1  yard  of  cloth  if  3^- 
yards  cost  25-f  c.  ?  Ans.  7^  c. 

37.™  If  a  man  travels  4  miles  in  1  hour,  how  far  will 
he  go  in  3|  hours? 

38.  If  a  horse  travels  6^  miles  in  1  hour,  how  far 
will  he  go  in  5^  hours?  Ans.  33J  miles. 

• 

203.  QUESTIONS  FOR  REVIEW. 

What  is:  1.  A  fractional  unit?  2.  A  fraction?  3.  The  terms? 
4.  The  denominator  ?  5.  The  numerator  ?  6.  A  proper  fraction  ? 
7.  An  improper  fraction  ?  8.  A  mixed  number  ?  9.  The  value 
of  a  fraction?  10.  A  compound  fraction?  11.  The  reciprocal  of  a 
fraction  ? 

How  may  a  fractional  unit  be  :  1.  Multiplied  by  a  whole  num- 
ber ?  2.  Divided  by  a  whole  number  ? 

Why  is  the  value  of  a  fraction  not  changed  by :  1.  Multi- 
plying both  terms  by  the  same  number  ?  2.  Dividing  both  terms 
by  the  same  number? 

What  is  reduction  of  fractions  ?  Give  the  rule  for  reducing  :  1. 
A  fraction  to  its  lowest  terms.  2.  A  whole  number  to  a  fraction. 
3.  A  fraction  to  higher  terms.  4.  A  mixed  number  to  an  improper 
fraction.  5.  An  improper  fraction  to  a  whole  or  mixed  number. 
6.  Compound  fractions  to  simple  ones.  7.  Fractions  to  a  com- 
mon denominator.  8.  Fractions  to  their  least  common  denomin- 
ator. 

What  is  the  principle  of:  1.  Addition?  2.  Subtraction?  3. 
Multiplication?  4.  Division? 

What  is  the  rule  for:  1.  Addition?  2.  Subtraction?  3.  Mul- 
tiplication ?  4.  Division  ? 

How  do  we  find  how  often  a  fraction  is  contained  in  1  ?  In  di- 
vision of  fractions,  why  do  we  invert  the  divisor  ? 


DECIMAL  FRACTIONS. 


INDUCTIVE  EXERCISES. 

204.  If  an  orange,  an  apple,  a  number,  or  a  bar  of 
soap  be  divided  into  ten  equal  parts,  what  is  one  of 
the  parts  called?  Ans.  1 
tenth.  What  are  two  of  the 
parts  called?  Three  of  the 
parts?  Four?  Five? 

If,  now,  each  of  these  tenths  be  divided  into  ten  equal 
parts,  what  is  one  of  the  parts  called?  Ans.  1  hun- 
dredth. What  are  two  of  the  parts  called?  Three  of 
the  parts?  Four?  Five? 

If,  now,  each  of.  these  hundredths  be  divided  into 
ten  equal  parts,  what  is  one  of  the  parts  called  ? 

Ans.  \  thousandth. 

In  the  number  327,  what  is  the  unit  of  7  ?  Ans.  one. 
See  Art.  31.  What  is  the  unit  of  2?  Of  3?  Is  the 
unit  of  2  one-tenth  of  the  unit  of  3?  Is  the  unit  of  7 
one-tenth  of  the  unit  of  2? 

If,  now,  we  write  other  figures  after  7,  thus:  327568, 
will  the  unit  of  5  be  1  tenth  of  the  unit  of  7?  Will 
the  unit  of  6  be  1  tenth  of  the  unit  of  5?  Will  the 
unit  of  8  be  1  tenth  of  the  unit  of  6  ? 

Placing  a  point  ( . )  after  7,  thus :  327.568,  indicates 
that  its  unit  is  one  j  what,  then,  is  the  unit  of  5  ?  Of 
6?  Of  8? 

(164) 


Decimal  Fractions. 


165 


1.  What,  then,  is  denoted  by  327.568? 

Ans.  327  and  5   tenths  6   hundredths   8  thousandths. 
Or,  327  and  £>  +  T£o  +  T<TO o  =  327  and  T4f&»  or  327 
and  568  thousandths. 

2.  What  is  denoted  by  125.34? 

Ans.  125  and  34  hundredths. 
3  What  is  denoted  by  804.05? 

Ans.  804  and  5  hundredths. 
4.  What  is  denoted  by  .125?  Ans.  125  thousandths. 

DEFINITIONS. 

205,  A  Decimal   Fraction  is  a  fraction  whose  denom- 
inator is  always  10,  100,  1000,  etc.,  or  1  with  O's  annexed. 

Decimal  Fractions  are  usually  called  decimals. 

206.  A  Decimal  is  usually  expressed  by  writing  only 
the  numerator  with  a  period  and   a  certain   number  of 
O's  before  it ;  the  denominator  being  understood,  but  not 
written. 

Thus  : 

l.    A    is  written  .1 


2.  A 

3<  TT7 

4.  T" 

5 


a 


«•  -fft 


TOOT  H 

5  written  .007 

27 
1000 

" 

.027 

134 
TFOU" 

a 

.134 

5 

ll 

.0005 

1  0000 

304 
10000 

" 

.0304 

2  5 

TTTrTTTTTriT 

a 

.000025 

The  period  placed  before  decimals  is  called  the 
decimal  point;  it  is  also  called  separatrix,  as  it  separates 
the  decimal  from  the  whole  number. 

207.  A  Mixed  Number  is  a  number  formed  of  a  whole 
number  and  a  decimal,  as  4.25,  of  which  4  is  the  whole 
number,  and  .25,  or  25  hundredths,  is  the  decimal. 


166  Intermediate  Arithmetic. 


PRINCIPLES. 

208.  1°.  Decimals   decrease    from   left   to   right   in   a 
tenfold  ratio,  just  as  whole  numbers  do. 

Hence,  the  value  of  a  decimal  figure   depends   upon   its  dis- 
tance from  the  decimal  point.     Thus,  .07  is  T£<y  ;  .007  is 


2°.  The  first  place  on  the  right  of  the  decimal  point 
is  that  of  tenths;  the  second  place  hundredths;  the  third, 
thousandths ;  the  fourth,  ten-thousandths ;  the  fifth,  hun- 
dred-thousandths;  the  sixth,  millionths,  etc. 

DECIMAL  NUMERATION  TABLE. 

02 

•  i  *  .4.3 

H3  .5  -JS  OQ 

S  .  O  j§        U3         S 

C3  «J  .  PL|  "S  ±j  02 

§     -2     -S  ^     "S      1 


3      I     1      ,      *     1     I     I      1     3 


^     §     1      §     |     |     1     §     §     fl 

IgS^Pft^WgS 

5432      1.       1234 

3°.  The  denominator  (understood)  of  a  decimal  is  1 
with  as  many  O's  annexed  as  there  are  figures  or  places 
in  the  decimal. 


What  is  the  denominator  of: 


1.  .6?        Ans.  10. 

2.  .43?      Ans.  100. 

3.  .07?     Ans.  100. 


4.  .027? 

5.  .009? 

6.  .0005? 


7.  .0325? 

8.  .53207? 

9.  .00506? 


209.  Expressing  decimals  as  common  fractions. 
1.  Express  .0205  as  a  common  fraction. 

Writing  the  denominator,  we  have  f  §f  {$.     Now,  removing  the 
point  and  the  cipher  following  it,  we  obtain  T 


Decimal  Fractions. 


1G7 


Express  as  a  common  fraction  : 


2.  .0027. 

3.  3.005. 
5.  75.04. 


A  A-)  o  2_7_ 

.xT./t'O*        -i  "TV  A  ()  (T* 

Ans.  o  j  0  0  0 . 


5.  .00325. 
6.  54.0107. 
7.  84.605. 

yins.  -;-• 
Ans.    ? 
Ans.    ? 

325 

roT)f><ro- 


210.  EXERCISES  IN  NUMERATION  AND  NOTATION. 


1.  Read  .025. 

2.  Read  13.000325. 


Ans.  25  thousandths. 
Ans.  13  and  325  millionths. 


Hence,  the 

RULE. — Decimals  are  read  as  they  would  be  if  expressed 
as  common  fractions. 

NOTE. — In  case  of  mixed  numbers,  the  decimal  point  is  called 
and,  which  is  emphasized  and  dwelt  upon,  to  indicate  the  termi- 
nation of  the  whole  number,  and  the  beginning  of  the  decimal. 

Read  : 

3.  .0605.     Ans.  605  ten-thousandths. 

4.  24.12.     Ans.  24  and  12  hundredths. 

5.  3.017.     Ans.  3  and  17  thousandths. 

6.  125.01.  Ans.  125  and  1  hundredth. 

7.  .004.       Ans.  4  thousandths. 


8.  25.036. 

9.  20.03. 

10.  101.101. 

11.  73.00576. 

12.  184.37504. 


Write  the  following  in  the  decimal  notation : 


13.  8  tenths. 

14.  24  hundredths. 

15.  7  hundredths. 

16.  5  and  16  thousandths. 

17.  12  and  3  hundredths. 


18.  3  and  2  tenths. 

19.  15  and  2  hundredths. 

20.  7  thousandths. 

21.  325  ten-thousandths. 

22.  4075  millionths. 


PRINCIPLES. 


T%  =  iW  --=  T7A°o0o,  etc. 


211,  Since  T7¥ 
We  have    .7=  .70=  .700=     .7000,  etc. 


168  Intermediate  Arithmetic. 

Hence, 

1°.  Annexing   one  or  more  ciphers  to  a  decimal  does  not 
alter  its  value. 

212.  Ten  times  25.78  is  257.8. 

For  10  times  2  tens  5  ones  7  tenths  8  hundredths  are  2  bunds. 
5  tens  7  ones  8  tenths,  or  257.8.     Hence, 

2°.   To  multiply  a  decimal  by  10,  100,  1000,  etc.,  we  re- 
move the  decimal  point  one,  two,  three,  etc.,  places  to  the  right. 

213.  Reversing  the  foregoing  principle,  we  have 

3°.   To  divide  a  decimal  by  10,  100,  1000,  etc.,  we  remove 
the  decimal  point  one,  two,  three,  etc. ,  places  to  the  left. 

EXERCISES. 

214.  Which  is  the  more : 


1.  .27  or  .2700? 

2.  4.6  or  4.60? 

Multiply  : 

5.  4.35  by  10.       Ans.  43.5. 

6.  75.07  by  1000.  Ans.  75070. 

7.  .0374  by  100.   Ans.  3.74. 

Divide  : 

11.  3.5  by  100.    Ans.  ,035. 

12.  64.5  by  10.   Ans.  6.45. 

13.  13  by  1000.  Ans.  .013. 


3.  43.063  or  43.06300? 

4.  705.47  or  705.47000? 


8.  .00305  by  1000.   Ans.  ? 

9.  350.47  by  1000.   Ans.  ? 
10.  47.319  by  10000.  Ans.  ? 


14.  47.365  by  100.        Ans.  ? 

15.  .002576  by  1000.     Ans.  ? 

16.  32547.1  by  100000.4ns.? 


215.  To  reduce  a  decimal  to  a  common  fraction  in  its 
lowest  terms. 

1  Reduce  .125   to   a  common   fraction    in   its   lowest 
terms. 

OPERATION.— .125  =  :  TWo  =  sVo  =  ¥0  =  =  1- 


Decimal  Fractions.  169 


Hence,  the 


RULE. — Express  the  decimal  as  a  common  fraction,   and 
reduce  it  to  its  lowest  terms. 

Reduce  the  following   to   common   fractions   in  their 
lowest  terms  : 


^- 


2.  .75.  Ans. 

3.  8.25.  Ans.  8J. 

4.  .035.  Ans. 

5.  .0625.  ^4ns. 


T6- 


7.  63.6.  Ans.  ? 

8.  63.600.  Ans.  ? 

9.  47.3125.  ^n*.  ? 
10.  .001875.  ^TIS.  ? 

6.  34.375.  Ans.  34|. !      11.  11.3125.  ^s.  ? 

216.  To  reduce  common  fractions  to  decimals. 
1.  Reduce  f  to  a  decimal. 

EXPLANATION.  —  3  ==  30   tenths ;   30  tenths  -s-  8      OPERATION 
=  3   tenths   and   6   tenths    over.    6  tenths  ==60          Q\onnn 
hundredths;  60  hundredths -^8  ==7  hundredths 
and    4    hundredths    over.       4    hundredths  —  40  .375 

thousandths,    and    this  -=-8=5    thousandths. 
Hence,  the  answer  is  3  tenths,  7  hundredths,  5  thousandths  = 
.375. 

Hence,  the 

RULE. — Annex  ciphers  to  the  numerator,  divide  by  the 
denominator,  and  from  the  right  of  the  quotient  point  off  as 
many  decimal  figures  as  there  are  ciphers  annexed. 


Reduce  to  decimals : 
2.  f.  Ans.  .75. 


4 


3.  f.  Ans.  .625. 

4.  f.  Ans.  .428+. 

5.  51.  Ans.  5.5. 

6.  7J.  Am.  7.25. 


7.  9-|.  Ans.  9.6. 

8.  -,  Ans.  .03125. 


9.  13^.  Ans.  ? 

10.  J 

11.  44.  Ans.  ? 


170  Intermediate  Arithmetic.  • 

ADDITION  OF  DECIMALS. 

217.  Since  decimals  increase  and  decrease  regularly 
by  the  scale  of  ten,  they  are  evidently  added  like  whole 
numbers.  Hence,  the 

RuLE.--W/rite  the  numbers  so  that  points  shall  stand 
under  points,  tenths  under  tenths,  hundredths  under  hun- 
dredths,  etc.,  and  add  as  in  whole  numbers. 


OPERATION. 


5.75 


1.  Add  together  5.75,  16.263,  143098 

143.098,  and  .96. 


166.071  Arts. 
Add  together  : 

2.  53.246,  44.82,  706.4,  49.82,  and  .5.     Ans.     854.786. 

3.  58.07,  43.9,  .84,  .679,  and  9.3.  Ans.    112.789. 

4.  9.74,  16.07,  924.,  75.24,  and  879.        Ans.  1025.929. 

5.  170.,  309.6,  58.754,  3.7,  and  .0349.      Ans.  542.0889. 

6.  23  and  7  hund'ths,  5  and  9  tenths,  271  and  46 
thous'ths,  and  133  and  575  ten-thous'ths.  Ans.  433.0735. 

7.  27  hund'ths,  83  thous'ths,  984  thous'ths,  7  and  8 
hund'ths,  and  74  and  125  ten-thous'ths.   Ans.  82.4295. 

8.  43  and  9  tenths,  13  and  13  thous'ths,  61  hund'ths, 
5  and  17  thous'ths,  and  425  and  78  hund'ths.  Ans.  488.320. 

NOTE.—  In  the  following,  first  reduce  the  fractions  to  decimals 
by  Art.  216,  and  then  add. 

Find  the  sum  of: 

9.  f,  |,  and  £.  Ans.  1.875. 

10.  3i   9-|,  7i,  and  16|.  Ans.  36.55. 

11.  J,  5|,  16^,  and  ^.  Ans.  22.96875. 

12.  2531,  187J,  95f,  and  3-J.  Ans.  540.025. 


Decimal  Fractions.  171 

SUBTRACTION  OF  DECIMALS. 

218.  Decimals  are  subtracted  like  whole  numbers  for 
the  same  reason  that  they  are  added  as  such. 
Hence,  the 

RULE.— -TFKte  the  subtrahend  under  the  minuend,  so  that 
points  shall  stand  under  points,  tenths  under  tenths,  hun- 
dredths  under  hundredths,  etc.,  and  subtract  as  in  whole 
numbers. 

OPERATION.     34.046 

1.  From  34.046  take  9.78.  9.780 

24.266  Ans. 

2.  From  41.  take  23.35. 

OPERATION.      41.00 

We  write  ciphers  above  the  3  and  5,  as  23  35 

there  are  no  tenths  and  no  hund'ths  in  the 


minuend.  17. DO 

From : 

3.  34.16  take  17.75.  ,  Ans.  16.41. 

4.  9.6  take  7.035.  Ans.  2.565. 

5.  .7  take  .368.  Ans.  .332. 

6.  87.946  take  8.76.  Ans.  79.186. 

7.  1  take  .0001.  Ans.  0.9999. 

8.  93  hund'ths  take  175  thous'ths.  Ans.  .755. 

9.  23  and  5  tenths  take  12  and  176  millionths. 

Ans.  11.499824. 

10.  184  and  7  thous'ths  take  137  and  68  thous'ths. 

Ans.  46.939. 

11.  1  thousand  take  1  thousandths.  Ans.  999.999. 

12.  7  hunds.  take  7  hund'ths.  Ans.  699.93. 

NOTE. — In  the  following,  first  reduce  the  fractions  to  decimals, 
and  then  subtract. 


172  Intermediate  Arithmetic. 

What  is  the  value  of: 


13. 

8i-n 

? 

Ans 

. 

1.25. 

i 

6. 

"20"       "  3T  ' 

Ans.  .25625 

14. 

*--#? 

Ans. 

0 

.275. 

7. 

T6"  "  ~  TO" 

7  • 

Ans.  .17 

15. 

if--i? 

Ans. 

0.65. 

l 

8. 

24f—13 

2-Q 

? 

Ans.  11.325 

MULTIPLICATION  OF  DECIMALS. 

219.   1.  Multiply  .9  by  .07.  OPERATION. 

.9 

EXPLANATION. -- .9  =  T9o,  and  .07  =  T$o-.  ^, 

Now  by  Art.  194,  &  XTfa  =i*!fo=  -063. 

.063  Alls. 

Hence,  the 

RULE. — Multiply  as  in  whole  numbers,  and  in  the  pro- 
duct point  off  as  many  decimal  figures  from  the  right  as 
there  are  decimal  places  in  both  factors,  prefixing  ciphers 
when  necessary  to  supply  the  deficiency. 

EXERCISES. 

Multiply : 


2.  4.5  by  6.4.        Ans.  28.8. 

3.  7.08  by  3.2.    Ans.  22.656. 

4.  16.5  by  .008.    Ans.  0.132. 

5.  125.08  by  .25.  Ans.  31.27. 

6.  .0061  by  .05.  Ans.  000305. 


7.  122  by  .78. 

8.  3.25  by  16. 

9.  4.508  by  .24. 

10.  .1806  by  5.4. 

11,  .0586  by  .75. 


12.  The  distance  around  a  circle  is  3.1416  times  the 
distance  through  it ;  how  far  is  it  around  a  circular 
garden  if  it  is  75  yards  through  it?  Ans.  235.62  yards. 

DIVISION  OF  DECIMALS. 

220.  Since  Division  is  the  reverse  of  Multiplication, 
by  reversing  the  rule  of  the  latter  we  get  the 

RULE. — Divide  as  in  whole  numbers,  and  in  the  quotient 


Decimal  Fractions.  173 

point  off  as  many  decimal  figures  from  the  right  as  the 
dicimal  places  of  the  dividend  exceed  those  of  the  divisor, 
prefixing  ciphers,  if  ncccssari/,  to  supply  the  deficiency 

NOTES. -- 1.  When  there  are  more  decimal  places  in  the  di- 
visor than  in  the  dividend,  make  them  equal  by  annexing  ci- 
phers to  the  dividend  before  dividing. 

II.  If  there  is  a  remainder,  ciphers  may  be  annexed  to  it  as 
decimals,  and  the  division  continued  at  pleasure. 

III.  When  there  is  a  remainder  at  the  close  of  the  operation, 
the  sign  -f  should  be  annexed  to  the  quotient  to  show  that  it 
is  not  complete. 

EXERCISES. 

Divide : 

1.  177.6  by  2.4.  Ans.  74. 

2.  62.5  by  .25.  Ans.  250. 

3.  8.84  by  3.4.  Ans.  2.6. 

4.  3.139  by  .43.  Ans.  7.3. 


9 


5.  283.25  by  2.5.  Ans. 

6.  .0639  by  .09.  Ans.  ? 

7.  45.625  by  12.5.  Ans.  ? 

8.  23421.  by  2.11.  Ans.  ? 


9.  42.81  by  .346.  Ans.  123.728  -f. 

10.  12.82561  by  3.01.  Ans.  4.261. 

11.  983  by  6.6.  Ans.  148.939 -f. 

221.  QUESTIONS  FOR  REVIEW 

What  is:   1.   A    Decimal   Fraction?     2.  The   decimal   point? 

3.  A  mixed  number? 

How  do  decimals  decrease  from  left  to  right?  On  what  does 
the  value  of  a  decimal  figure  depend?  The  first  place  on  the 
right  of  the  decimal  is  that  of  what  ?  The  second  place  on  the 
right?  The  third?  The  fourth?  The  fifth? 

Is  the  denominator  of  a  decimal  written?    Wrhat  is  it? 

How  are  decimals  read?    Repeat  the  three  principles,  p.  168. 

How  are:  1.  Decimals  reduced  to  fractions  in  their  lowest 
terms?  2.  Common  fractions  reduced  to  decimals? 

How  are  decimals:  1.  Added?    2.  Subtracted?    3.  Multiplied? 

4.  Divided? 


UNITED  STATES  MONEY. 


COINS. 


-/  tt I-  -KK' 

?,  --  svs 


GOLD  COINS.— Eagle,  half  eagle,  quarter  eagle,  three  dollar  piece, 
and  dollar. 


SILVER  COINS.  —Dollar,  half  dollar,  quarter  dollar,  dime. 


NICKEL  AND  BRONZE  COINS. — 5-cent,  3-cent,  and  l-cent  pieces. 

( 17-1 ) 


United  States  Money.  175 

222.  The  Currency  of  a  nation  means  its  money. 

223.  U.  S.  Money  is  the  legal  currency  of  the  United 
States.     Its  denominations  are  Eagles  (E.),  Dollars 
Dimes  (d. ),  Cents  (c. ),  and  Mills  (  m.  ). 


10  m.  —  1  c. 
10  c.    =1  d. 
10  d.   --=  $1. 
$10      =  1  E. 


TABLE. 

E.       $.        d.  c.  m. 

1  =  10  =  100  =  1000  —  10000 

1  =    10  ==    100  =    1000 

1  =      10  =      100 


224.  The  U.  S.  coins  are  gold,  silver,  nickel,  and  bronze. 
In  addition  to  the  coins,  p.  174,  is  the  double  eagle  (gold). 

The  weight  of  the  gold  dollar  is  25.8  grains,  and  that  of  the 
silver  dollar  is  412£  grains. 

225.  The  Dollar  is  the  Unit,  and  the  only  denomina- 
tions used  in  practice  are  dollars  and  cents.     Cents  are 
hundredths,  and  mills  thousandths  of  a  dollar. 

Hence,  dollars  are  written  with  the  sign  ( $ )  prefixed  to  them, 
and  the  point  (  . )  placed  after  them,  and  cents  and  mills  are 
written  in  the  hundredths  and  thousandths  places  respectively 
on  the  right  of  the  point. 

Thus,  we  write  12  dollars  23  cents  and  6  mills,  $12.236 ; 
12  dollars   4   cents  and  3  mills,  $12.043 ; 
12  dollars  and  7  mills,  $12.007. 
All  vacant  places  are  filled  with  O's. 

REDUCTION  OF  U.  S.  MONEY. 

MENTAL  EXERCISES. 

226.  How  many  cents  in  : 

l.l  dime?    5  d.  ?    7  d.  ?    10  d.  ?    23d.?    £  d.  ?   2Jd.? 

2.  1  dollar?     $3?     $8?     $10?     $67?     H?     H?     $t? 

3.  10    mills?      20   m.?      40  m.  ?      100  m.  ?      130  m.  ? 


176  Intermediate  Arithmetic. 

How  many  dimes  in  : 

4.  1  dollar?    $7?    $9?    $45?    $1?    $5J?    $T3o?    $6TV? 

5.  1   cent?      30  c.  ?      170  c.  ?      43   c.  ?      (Ans.  4.3  c.) 
75  c.?     112  c.?    5  c.? 

How  many  mills  in  : 

6.  1   cent?     7  c.?     31   c.  ?     125   c.  ?     83  c.  ?     £  c.  ? 
4}  c.  ?     10  c.  ? 

7.  1   dime?     2  d.?     5   d.?      17  d.  ?     |  d.?     2J  d.  ? 
TV  d.?    A  d.? 

8.  1  dollar?    $3?    $12?    $J?    $TV?    $T9o?    H? 


227.  CASE  I.  —  From  a  higher  to  a  lower  denomination. 

I.  Reduce  $5  to  cents;  also  to  mills. 

In  $1  there  are  100  c.  ;  hence,  OPERATION. 

in  $5  there  are  5  times  100  c.  =       5  x/  JQQ  c  __  PJQQ  c 

5°°TCe^;  mnn        i,  5  X  1000  in.  =  5000  m. 

In  $1  there  are  1000  m.  ;  hence, 

in  $5  there  are  5  times  1000  m.  =  5000  mills. 

Hence,  the 

RULE.  —  I.   To  reduce  dollars  to  cents,  multiply  by  100  or 
annex  two  ciphers. 

II.  To  reduce  dollars  to  mills,  multiply  by  1000,  or  annex 
three  ciphers. 

EXERCISES. 

Reduce  : 


2.  $7  to  cents.        Ans.  700  c. 

3.  $19  to  mills.  Ans.  19000  m. 

4.  $125  to  mills.  Ans.  ? 

How  many : 

8.  Cents  in  $J?  Ans.  50. 

9.  Mills  in  $f?  Ans.  750. 
10.  Cents  in  $f  ?  Ans.   ? 


5.  $11  to  cents.  Ans.  ? 

6.  $162  to  mills.         Ans.  ? 

7.  $3274  to  cents.        Ans.  ? 


11.  Mills  in  $|?     Ans.  625. 

12.  Cents  in  $i|? 

13.  Mills  in  $H?  Ans.   ? 


United  States  Money.  177 

228.  CASE  II. — From  a  lower  to  a  higher  denomination. 

RULE. —  To  reduce  cents  or  mills  to  dollars,  reverse  the 
rule  in  Art.  227,  and  divide  by  100  or  1000,  or  point  off  turn 
or  three  figures  from  the  right  for  decimals. 

EXERCISES. 

1.  Reduce  375  cents  to  dollars.  Ans.  83.75. 

2.  Reduce  4261  mills  to  dollars.  Ans.  84.261. 


How  many  dollars  in  : 

3.  37  cents?       Ans.  .37. 

4.  421  mills?      Ans.  .421. 

5.  679  cents?     Ans.  ? 


6.  8743  mills?         Ans.  ? 

7.  75  cents?  Ans.  f. 

8.  875  mills?          Ans.  £. 


EXERCISES  IN  U.  S.  MONEY. 

229.  Addition,    Subtraction,    Multiplication,   and    Di- 
vision of  U.  S.  Money  are  evidently  performed  accord- 
ing to  the  rules  of  decimal  fractions. 

230.  i.  Add  together  8473.43,  8530.75,  8645.29,  8432.19, 
85663.25.  Ans.  87744.91. 

2.  What  is  the  number  whose  c  parts  are  845,  837.50, 
818.75,  845.25,  and  812.25?  Ans.  8158.75. 

3.  A  farmer  sold   a  lot  of  cotton   for   81235,  a  lot  of 
corn  for  8526.35,  a  load  of  peas  for  837-245,  a  yoke  of 
oxen  for  842,   and    a    heifer   for   818.235 ;    what  was  the 
total  amount?  Ans.  81858.83. 

4.  From  8327.59  take  8163.75.  An*.  $163.84. 

5.  Find  the  difference  between  8545  and  8275.25. 

Ans.  8269.7"'. 

6.  A   owes   B   8325.05   and   B  owes   A  8284.785;   how 
should  they  settle?  Ans.   A  should  pay  B  $40.265. 

N.  I.— 12. 


178  Intermediate  Arithmetic. 

7.  The    multiplicand    is    $434.25,   the    multiplier    is 
3.075;  what  is  the  product?  Ans.  $1335.31875. 

8.  What  is  the  value  of  9  bales  of  cotton,  averaging 
450.6  pounds  a  bale,  and  worth   8.5   cents  per  pound? 

Ans.  $344.709. 

9.  A  farmer  sold  8.6  barrels  of  syrup,  averaging  31.23 
gallons  a  barrel,  at  62.5  cents  per  gallon ;   what  did  he 
receive  for  all?  Ans.  $167.86+. 

10.  What  is  the  quotient  of  $984.15  by  243? 

Ans.  $4.05. 

11.  The  dividend  is  $63.25,  the  divisor  2.7;   what  is 
the  quotient?  Ans.  $23.42+. 

12.  A  farmer  sold  6.4  acres  for  $148.992;   how  much 
was  that  per  acre  ?  Ans.  ? 

13.  Among  how  many  persons  can  $197.4  be  distrib- 
uted if  each  person  receives  $1.128?  Ans.  175. 

14.  A  merchant  bought  125  barrels  of  apples  at  $3.50, 
and  sold  40  barrels  at  $3.25,  and  the  remainder  at  $4.10 
a  barrel ;  how  much  did  he  gain  or  lose  by  the  opera- 
tion ?  Ans.  $41   gain. 

15.  A   farmer  bought   160   hogs  at  $4.25   a  head,   18 
sheep  at  $6.50  a  head,  two  wagons  at  $87.45  apiece,  and 
paid  in  cash  $604.25;  how  much  did  he  then  owe? 

Ans.  $367.65. 


ACCOUNTS  AND  BILLS. 

231.  A  Debt  is  money,  goods,  or  services  due  from  one 
party  to  another.      A  debtor  is  a  person  who  owes  a  debt, 
a  creditor  one  to  whom  a  debt  is  due. 

232.  A  Bill  of  Goods  is  a  written  statement  given  by 


Accounts  and  Bills.  179 

the  seller  to  the  buyer,  containing  the  date  of  the  pur- 
chase, the  names  of  the  buyer  and  seller,  a  list  of  the 
goods  bought,  with  their  prices,  and  the  total  amount. 

An  Item  is  any  article  in  the  bill ;  extending  an  item 
is  finding  its  cost,  and  the  footing  is  the  entire  cost  of 
all  the  items  in  a  bill. 

233.  An  Account  is  a  written  statement  of  debits  and 
credits  between  two  parties. 

When  a  bill  or  account  is  paid,  the  creditor  should 
write  at  the  bottom  of  the  same  :  Received  payment,  and 
after  it,  his  name. 

The  symbol  @  stands  for  at. 


BILLS. 

334.  Extend  the  items  and  find  the  footings  of  the 
following  bills : 

(1) 

LOUISVILLE,  June  £3,  1885. 

Dr.   W.  M.  Baker, 

Bought  of  George  Coleman. 


5  yards  broadcloth  @  $3.25. . 
3  yards  cambric  @»  $.12|.... 

3  dozen  buttons  @  $.15 

6  Skeins  sewing  silk  (''•  $.0tl£ 

4  yards  wadding  @,  $.08  .... 

Amount. 


Received  payment, 

George  Coleman. 


180 


Intermediate  Arithmetic. 


(2) 

BATON  ROUGE,  June  16,  1SS5. 
Mr.  J.  R.  Holmes, 

Bought  of  Win.  Garig  &  Co. 


86  pounds  coffee  @  10^  c. . 

38  pounds  tea  @  85  c , 

63  gallons  molasses  @  37|  c, 

125  pounds  rice  @  7  c , 

75  pounds  starch  @  4  c 

56  pounds  bar-soap  @  5£  c. , 

Amount. 


79 


43 


Received  payment, 

Wm.  Garig  &  Co. 

235.  QUESTIONS  FOR  REVIEW. 

What  is  U.  S.  money?  What  are  the  denominations?  Repeat 
the  table. 

What  are  :  1 .  The  gold  coins  ?  2.  The  silver  coins  ?  3.  The 
nickel  coins?  4.  The  bronze  coins? 

What  is  the  unit?  What  denominations  only  are  used  in  prac- 
tice? How  are  :  1.  Dollars  expressed  ?  2.  Cents  expressed?  3. 
Mills  expressed? 

How  are  dollars  reduced:  1.  To  cents?    2.  To  mills? 

How  are :  1.  Cents  reduced  to  dollars  ?  2.  Mills  reduced  to 
dollars  ? 

How  are  the  operations  of  Addition,  Subtraction,  Multiplication, 
and  Division  of  U.  8.  money  performed? 

What  is:  1.  A  debt?  2.  A  debtor?  3.  A  creditor  ?  4.  A  bill 
of  goods?  5.  An  item?  6.  An  account? 

What  is  meant  by  :  1.  Extending  an  item?  2.  Finding  the 
footing  of  a  bill  ? 

When  a  bill  or  an  account  is  paid,  what  should  be  done  ? 


COMPOUND  NUMBERS. 


DEFINITIONS. 

236  A  Measure  is  a  standard  unit  of  quantity  by 
which  similar  quantities  are  measured,  and  their 
amounts  or  values  estimated ;  as  1  pound,  1  hour,  1 
dime,  etc. 

237.  A    Simple    Number  is    an  amount  expressed  in 
terms  of  one  measure ;  as  5  feet,  9  pounds. 

238.  A  Compound  Number  is  an  amount  expressed  in 
terms  of  different,  but  similar,  measures ;  as  5  yards  2 
feet  7  inches,  9  weeks  5  days.     7  yards  3  days  is  not  a 
compound  number,  as  the  measures,  yard  and  day,  are 
not  similar  things. 

LINEAR  MEASURE. 

239.  The  Measures    used    in   measuring  distances,  as 
length,  width,  and  height,  are  the  mile  (mi.),  the  chain 
(ch.),  the  rod  (rd.),  the  yard  (yd.),  the  foot  (ft),  and  the 
inch  (in.). 

TABLE. 

12  in.  =  1  ft*  4  rd.  =  1  ch. 

3  ft.  =  1  yd.  80  ch.  =  1  mi. 

5^  yd.  =  1  rd. 


*  The  table  is  read :  12  inches  are  1  foot,  3  feet  are  1  yard,  etc. 

(181) 


182  Intermediate  Arithmetic. 

MENTAL  EXERC/SES, 

240.   l.  How  many  inches  in  2  feet?    5  feet?    7  feet? 
£  foet?    i  foot? 

2.  How  many  feet  in  36  inches  ?     60  in.  ?      108  in.  ? 
30  in.  ?     6  in.  ? 

3.  How  many  feet  in  2  yards  ?     9  yds.  ?     20  yds.  ?     | 
yd.?    iyd.? 

4.  How  many   yards   in    9   feet?      11  feet?      10  rd.  ? 
36  in.  ?     72  in.  ? 

5.  How  many  rods  in  3  chains.  ?     5  ch.?     7  ch.?    \ 
ch.?     11  yd.? 

6.  How  many    chains  in   2   miles  ?     ^  mi.?     J  mi.  ? 


WRITTEN  EXERCISES. 

241.  Reduction   is  the  process  of  changing  the  meas- 
ures of  numbers  without  changing  their  amounts.      It 
is  of  two  kinds,  viz  :  Descending  and  Ascending. 

242.  I.    Reduction    Descending    is    changing    from    a 
higher  to  a  lower  measure. 

1.  Reduce  8  yd.  2  ft.  7  in.  to  inches. 

OPERATION. 

EXPLANATION.—  Since  1  yd.  =3  ft.,  we  mul-         g  y(j  2  ft  7  in. 
tiply  8  by  3  to  reduce  the  yds.  to  ft.,  and  add         o  ' 
in  the  2  ft.,  making  26  ft.     Again,  since  1  ft.    — 
=  12  in.,  we  multiply  26  by  12  to  reduce  the 
ft.  to  in.,  and  add  in  the  7  in.,  making  319       12 
inches. 


243.  Hence,  for  reduction  descending,  we  have  the 

RULE  I.  —  Multiply  the  number  of  the  highest  measure  by 
the  number  required  of  the  next  lower  measure  to  make  one 


Compound  Numbers.  183 

of  the   higher,  and  to   the  product   add   th-e  number  of  the 
lower  measure,  if  any. 

II.  Proceed  in  like  manner  with  the  result,  and  so  con- 
tinue until  the  required  measure  is  reached. 

244.  II.  Reduction  Ascending  is  changing  from  a  lower 
to  a  higher  measure. 

2.  Reduce  319  in.  to  yards. 

EXPLANATION.— Since  1  ft.  ==12  in.,  we  di-  OPERATION. 

vide  the  319  in.  by  12  to  reduce  to  ft.,  and  ob-  IO^QI q  jn 
tain  26  ft.  and  7  in.  over.   Again,  since  1  yd.= 
3  ft.,  we  divide  26  ft.  by  3  to  reduce  to  yd.,       3)26_ft.  7  in. 
and  get  8  yd.  and  2  ft.  over.     We  thus  ob-  8  yd.  2  ft. 

tain  8  yd.  2  ft.  7  in.,  Ans. 

245.  Hence,  for  reduction  ascending,  we  have  the 

RULE  I. — Divide  the  number  by  the  number  required  of 
its  measure  to  make  one  of  the  next  higher. 

II.  In  the  same  manner  divide  the  quotient,  and  so  on, 
until  the  required  measure  is  reached.  The  last  quotient 
with  the  remainders  annexed,  will  be  the  required  result. 

Reduce  : 

3.  10  yds.  2  ft.  4  in.  to  inches.  Ans.  388  in. 

4.  139  in.  to  yards.  Ans.  3  yd.  2  ft.  7  in. 

5.  6  mi.  22  ch.  2  rd.  to  rods.  Ans.  2010  rd. 

6.  8  ch.  2  rd.  3  yd.  to  yards.  Ans.  190  yd. 

7.  763  rd.  to  miles.  Ans.  2  mi.  30  ch.  3  rd. 

8.  16  rd.  2  ft.  to  feet.  Ans.  266  ft. 

9.  8375  in.  to  rd.  Ans.    ? 
10.  20  mi.  12  ch.  2  yd.  5  in.  to  inches.  Ans.  ? 

SQUARE  MEASURE. 

246.  A   Surface  is   that  which   has  only  two   dimen- 


184 


Intermediate  Arithmetic. 


sions :  length  and  width  ;  as  the  face  of  a  black-board,  or 
the  surface  of  a  floor  or  slate. 

All  surfaces  are  measured  by  squares,  like  those  on  a  chess- 
board, or  like  this  figure-: 

A  square  inch  is  a  surface  1  inch 
long  and  1  inch  wide. 

A  square  foot  is  a  surface  1  foot 
long  and  1  foot  wide. 

The  measures  used  in  measur- 
ing surfaces  are  the  square  mile 
(sq.  mi,),  the  acre  (A.),  the  square 


ONE 

SQUARE 

FOOT 


A  SQUARE  YARD. 


chain  (sq.  ch,),  the  square  rod 
(  sq.  rd. ),  the  square  yard  (  sq.  yd.), 
the  square  foot  (sq.  ft.),  and  the  squre  inch  (sq,  in.) 


TABLE. 


16  sq.  rd.  =  1  sq.  ch. 
10  sq.  ch.  —  1  A. 
640  A.          =  1  sq.  mi, 

Measures  sometimes  used :  1  sq.  rd.  =  1  perch  or  pole 
(P.) ;  40  P.  =  1  rood  (E.)  ;  4  R.  =  1  acre. 


144  sq.  in.   =  1  sq.  ft. 
9  sq.  ft,    =  1  sq.  yd. 
sq.  yd.  =  1  sq.  rd. 


MENTAL  EXERCISES. 

247.   1.  How   many   sq.   in.    in    2  sq.  ft.?    5    sq.  ft.? 
|  sq.  ft.  ?     J  sq.  ft.  ? 

2.  How   many   sq.   ft.  in    144   sq.   in.?     432   sq.  in.? 
72  sq.  in.  ?     36  sq.  in.  ? 

3.  How   many   sq.  ft.  in  3  sq.   yd.  ?     9  sq.   yd.  ?    20 
sq.  yd.  ?    £  sq.  yd.  ? 

4.  How  many  sq.  rd.  in  5  sq.  ch.  ?     7  sq.  ch.  ?    %  sq. 
ch.  ?     £  sq.  ch.  ? 

5.  How  many  acres  in  40  sq.  ch.  ?     100  sq.   ch.  ?     75 
sq.  ch.  ?     2  sq.  ch.  ? 


Solid  Measure. 


185 


WRITTEN  EXERCISES. 


248.  Reduce: 


1.  9  sq.  yd.  7  sq.  ft.  to  square  feet.          Ans.  88  sq.  ft. 

2.  93  sq.  ft.  to  square  yards.     Ans.  10  sq.  yd.  3  sq.  ft. 

3.  84  sq.  rd.  12  sq.  yd.  6  sq.  ft.  to  square  feet. 

Ans.  22983  sq.  ft. 

4.  583  sq.  rd.  to  acres.       Ans.  3  A.  6  sq.  ch.  7  sq.  rd. 

5.  17  A.  3  R.  15  P.  to  poles  or  perches.     Ans.  2855  P. 

6.  13573  sq.  ch.  to  square  miles. 

Am.  2  sq.  mi.  77  A.  3  sq.  ch. 

7.  16725  sq.  in.  to  square  yards. 

Ans.  12  sq.  yd.  8  sq.  ft.  21  sq.  in. 

8.  16  sq.  rd.  5  sq.  ft.  to  square  inches.  Ans.  ? 

9.  11  A.  8  sq.  ch.  3  sq.  yd.  to  square  feet.  Ans.  ? 
10.  1  sq.  mi.  to  inches.                                              Ans.  ? 


SOLID  OR  CUBIC  MEASURE. 

249.  A  Volume  or  Solid  is  that  which  has  three  di- 
mensions :  length,  width  and  height ;  as  a  box,  a  room, 
or  a  book. 

All  volumes  are  measured  by  cubes  like  this  figure : 

A  cubic  inch  is  a  volume  1  in.  long, 
1  in.  wide,  1  in.  high. 

A  cubic  foot  is  a  volume  1  ft.  long, 
1  ft.  wide,  1  ft.  high. 

The  measures  used  in  measuring 
volumes  are  the  cord  (c. ),  the  cubic 
yard  (cu.  yd. ),  the  cubic  foot  (  cu.  ft.), 
and  the  cubic  inch  ( cu.  in. )  A  CUBIC  YARD. 


186  Intermediate  Arithmetic. 

TABLE. 

* 

1728  cu.  in.  =  1  cu.  ft.  128  cu.  ft.  =  1  c. 

27  cu.  ft.  =  1  cu.  yd. 

A  cord  is  used  for  measuring  wood.  A  pile  of  wood  8  feet 
long,  4  feet  wide,  and  4  feet  high  is  a  cord.  One  foot  in  length 
of  this  pile,  or  16  cu.  ft.,  is  a  cord  foot. 

MENTAL  EXERCISES. 

250.  l.  How   many  cu.  ft.  in  2  cu.  yd.?    5  cu.  yd.? 
^  cu.  yd.  ?    |  cu.  yd.  ? 

2.  How  many  cu.  yd.  in  81  cu.  ft.  ?  270  cu.  ft.  ?  9 
cu.  ft.  ?  3  cu.  ft.  ? 

WRITTEN  EXERCISES. 

251.  Reduce: 

1.  13  cu.  ft.  to  cubic  inches.  Ans.  22464  cu.  in. 

2.  25  cu.  ft.  524  cu.  in.  to  cubic  inches. 

Ans.  43724  cu.  in. 

3.  2379  cu.  in.  to  cu.  ft.  Ans.  1  cu.  ft.  651  cu.  in. 

4.  9  cu.  yd.  123  cu.  in.  to  cubic  inches. 

Ans.  420027  cu.  in. 

5.  1274  cu.  ft.  to  cords.  Ans.  ? 


MEASURES  OF  CAPACITY. 

Capacity  means  amount  of  bulk  or  space. 

I.  LIQUID  MEASURE. 

252.  The  Measures  used  in  measuring  liquids,  such  as 
water,  milk,  oil,  whisky,  etc.,  are  the  hogshead  (hhd.),  the 


Measures  of  Capacity. 


187 


barrel  (bar.  or  bbl.),  the  gallon  (gal.),  the  quart  (qt.),  the 
pint  (pt.),  and  the  gill  (gi.)- 


Gi.    PT.     QT. 


GAL. 


BAR. 


TABLE. 


HHD. 


4  gi.    =1  pt. 

2  pts.  =  1  qt. 
4  qts.  =  1  gal. 

One  gallon  contains  231  cu.  inches. 


311  gal. 
63  gal. 


1  bar. 
1  hhd. 


MENTAL  EXERCISES. 

253.  How  many  : 

1.  Gills  in  2  pt.?    5  pt.  ?     1  qt.  ?    3  qt.  ?     1  gal.?    2 
gal.  ?    i  gal.  ? 

2.  Pints  in  8  gi.  ?     32  gi.  ?     2  qt.  ?     7  qt.  ?     £  qt.  ?     3 
gal.  ?    |  gal.  ? 

3.  Quarts  in  4  pt.  ?     20  pt.  ?     2  gal.  ?     10  gal.  ?    £  gal  ? 
J  gal.  ?     32  gi.  ? 

4.  Gallons   in  1   bar.  ?    2  bar.  ?     2  hhd.  ?     20  qt.     56 
pt.  ?     64  gi  ?    £  hhd.  ? 


WRITTEN  EXERCISES. 


254.  Reduce: 


1.  7  gal.  2  qt.  1  pt.  to  pints.  Ans.  61  pt. 

2.  1039  gi.  to  gallons.     Ans.  32  gal.  1  qt.  1  pt.  3  gi. 

3.  32  gal.  2  qt.  1  pt.  to  gills.  Ans.   1044  gills. 


188 


Intermediate  Arithmetic. 


4.  367  pt.  to  gallons.  Ans.  45  gal.  3  qt.  1  pt. 

5.  4625  qt.  to  hogsheads.  Ans.  18  hhd.  22  gal.  1  qt. 

6.  3  hhd.  17  gal.  1  pt.  to  gills.                Am.  6596  gi. 

7.  10  bar.  3  gi.  to  gills.  Ans.  10083  gi. 

8.  3  hhd.  1  bar.  16J  gal.  to  pints.  Ans.  1896  pt. 

9.  1024  pt.  to  barrels.  Ans.  ? 
10.  1  hhd.  to  gills.  Ans.  ? 

II.  DRY  MEASURE. 

255.  The  measures  used  in  measuring  quantities  that 
are  not  liquid,  such  as  grain,  potatoes,  coal,  etc.,  are 
the  bushel  (bu.),  the  peck  (pk.),  the  quart  (qt.),  and  the 
pint  (pt,). 


PT.      QT. 


Bu. 


TABLE. 


2  pt.  =  1  qt.        8  qt.  =  1  pk.        4  pk.  =     1  bu. 

One  bushel  contains  2150.4  cu.  inches;  hence,  one  gallon,  Dry 
Measure,  contains  268.8  cu.  inches. 

MENTAL  EXERCISES. 

256.  How  many : 

l.  Pints   in   2   qt.?    7   qt.?    |  qt.  ?     1   pk.  ?    5  pk.  ? 


2.  Quarts  in  8  pt.  ?    20  pt.  ?     11  pt.  ?     2  pk.  ?     7  pk.? 
ipk.? 

3.  Pecks  in  16  qt.  ?    36  qt.  ?     16  pt.  ?    48  pt.  ?    2  bu.  ? 
t  bu.? 


Measures  of  Weiyht. 

WRITTEN  EXERCISES. 
257.  Reduce: 

1.  2  pk.  3  qt.  1  pt.  to  pints.  Ans.  39  pt. 

2.  10  bu.  3  pk.  7  qt.  to  quarts.  Ans.  351  qt. 

3.  57  pt.  to  pecks.                            Ans.  3  pk.  4  qt.  1  pt. 

4.  195  qt.  to  bushels.  Ans.  6  bu.  3  qt, 

5.  34  bu.  5  qt.  to  pints.  Ans.  ? 

6.  1765  pt.  to  bushels.  Ans.  ? 

7.  21  bu.  3  pk.  5  qt.  1  pt.  to  pints.  Ans.   ? 


MEASURES  OF  WEIGHT. 

I.  TROY  WEIGHT. 

258.  The  measures  used  in  weighing  precious  stones, 
gold,  silver,  etc.,  are  the  pound  (lb.),  the  ounce  (oz.),  the 
pennyweight  (pwt.),  and  the  grain  (gr.). 


GR.  PWT.  Oz. 

TABLE. 
24  gr.       1  pwt.        20  pwt.  :     1  oz.       12  oz.  =     1  lb. 

MENTAL  EXERCISES. 

259.  How  many  : 

1.  Grains  in  2  pwt.?     5  pwt.?    %  pwt.?    -J  pwt.?     J 
pwt.? 

2.  Pennyweights  in  72  gr.  ?     240  gr.  ?     3  oz.  ?    £  oz.  ? 
4  oz.? 

3.  Ounces  in  40  pwt.  ?    70  pwt.  ?    5  lb.  ?    £  lb.  ?   i  lb.  ? 


190  Intermediate  Arithmetic. 

WRITTEN  EXERCISES. 

260.  Reduce: 

1.  2  oz.  7  pwt.  19  gr.  to  grains.  Ans.  1147  gr. 

2.  5  Ib.  10  oz.  13  pwt.  to  pennyweights.  Ans.  1413  pwt. 

3.  8  Ib.  16  pwt.  to  grains.  Ans.  46364  gr. 

4.  245  pwt.  to  pounds.  Ans.  1  Ib.  5  pwt. 

5.  677  gr.  to  ounces.  Ans.  1  oz.  8  pwt.  5  gr. 

6.  8493  gr.  to  pounds.  Ans.     ? 

7.  205  oz.  12  gr.  to  grains.  Ans.     ? 

II.  AVOIRDUPOIS  WEIGHT. 

261.  The  measures  used  in  weighing  articles,  such  as 
hay,  cotton,  groceries,  etc.,  are  the  ton  (t.),  the  hundred- 
weight (cwt.),  the  pound  (Ib.),  the  ounce  (oz.),    and  the 
dram  (  dr.). 

TABLE. 


16  dr.  =  -- 1  oz. 
16  oz.  =1  Ib. 


100  Ib.  =  1  cwt. 
20  cwt.     :  1  t. 


MENTAL  EXERCISES. 

262.  How  many : 

1.  Drams   in   2   oz.  ?     4  oz.  ?     £  oz.  ?     \  oz.  ?     \  oz.  ? 

11    OZ.  ? 

2.  Which  is  the  lowest  measure?     The  next   lowest? 
etc. 

3.  Which  is  the  highest    measure?      The  next  high- 
est? etc. 

Reduce  : 

4.  3  t.  to  cwt.  ;  5  cwt.  to  Ib.  ;  10  Ib.  to  oz. ;  20  oz.  to  dr. 

5.  100  cwt.  to  t.;   700  Ib.  to  cwt.;  320  oz.  to  Ib. ;    160 
dr.  to  oz. 


Measures   of  Weight.  iyi 

WRITTEN  EXERCISES. 

263.  Reduce: 

1.  2  Ib.  5  oz.  10  dr.  to  drams.  Ans.  602  dr. 

2.  5  cwt.  80  Ib.  12  oz.  to  ounces.  Ans.  9292  oz. 

3.  7  t.  17  cwt.  50  Ib.  to  pounds.  Ans.  15750  Ib. 

4.  5285  pounds  to  tons.  Ans.  2  t.  12  cwt.  85  Ib. 

5.  4364  oz.  to  hundredweights.  Ans.  2  cwt.  72  Ib.  12  oz. 

6.  25607  dr.  to  pounds.  Ans.      ? 

7.  5  t.  10  cwt.  10  Ib.  12  oz.  to  drams.  Ans.      ? 

8.  512257  dr.  to  tons.  Ans.  1  t.  1  Ib.  1  dr. 

III.  APOTHECARIES'  WEIGHT. 

264.  The  measures  used  in  mixing  medicines  are  the 
pound  (Ib.  or  ft),  the  ounce  (oz.  or  §),  the  dram  (dr.  or  3), 
the  scruple  (scr.  or  9),  and  the  grain  (gr.). 


TABLE. 


20  gr.  -  -- 1  scr. 
3  scr.=  1  dr. 


8  dr.  =  1  oz. 
12  oz,  =  1  Ib. 


NOTE.  —  The  pound,  the  ounce,  and  the  grain  are  the  same 
measures  in  Troy  and  Apothecaries'  weight. 

MENTAL  EXERCISES. 

265.  How  many  : 

1.  Grains  in  3  scr.?     59?     7  scr.?      10  9  ?     £  scr.? 
i   ^  ? 

5    O  • 

2.  Which  is  the  lowest  measure?     The  next  lowest? 
etc. 

3.  Which  is  the  highest   measure?      The   next  high- 
est? etc. 


192  Intermediate  Arithmetic. 

WRITTEN  EXERCISES, 

266.  How  many: 

1.  Grains  in  5  oz.  6  dr.  2  scr.  10  gr.  ?     Ans.  2810  gr. 

2.  Scruples  in  3  Ib.  10  oz.  5  dr.  1  scr.  ?  Ans.  1120  scr. 

3.  Drams  in  5  ft>.  4  3  2  3?  Ans.  514  3. 

4.  Scruples  in  12  Ib.  7  dr.  ?  4ns.  3477  scr. 

5.  Pounds,  etc.,  in  99  dr.?  Ans.  1  Ib.  3  dr. 

6.  Ounces,  etc.,  in  167  scr.  ?         Ans.  6  oz.  7  dr.  2  scr. 

7.  Drams,  etc.,  in  583  gr.  ?  Ans.  9  dr.  2  scr.  3  gr. 

8.  Pounds,  etc.,  in  564307  grains?  Ans.     ? 

9.  Grains  in  5  Ib.  5  dr.  5  gr.  ?  Ans.     ? 


MEASURES  OP  MONEY. 

267. — I.  UNITED  STATES  MONEY. 

NOTE. — For  table  and  exercises  under  this  head,  see  United 
States  Money,  page  174. 

II.  ENGLISH  MONEY. 

268.  English  or  Sterling  money  is  the  currency  of 
Great  Britain.  The  denominations,  or  measures,  are  the 
pound  (£.),  the  shilling  (s.),  the  penny  (d.),  and  the 
farthing  (far.). 

TABLE. 

4  far.  =  1  d.  12  d.  =  1  s.  20  s.  =  1  £. 

NOTE.— A  florin  =  2  s. ;  a  guinea  =  21  s. ;  and  1  £==$4.84. 


Measures  of  Monet/.  193 

MENTAL  EXERCISES. 

269.  How  many : 

1.  Farthings  in  3d.?    5  d.  ?    |  d.  ?     i  d.  ? 

2.  Pence  in  20  far.  ?     30  far.  ?     2s.?     6s.?    £  s.  ? 

3.  Shillings  in  24  d.  ?     72  d.?     3  £?    1  £?    £  £? 

4.  Which  is  the  lowest  denomination?    The  next?  etc. 

5.  Which  is  the  highest  denomination?  The  next?  etc. 

WRITTEN  EXERCISES. 

270.  Reduce: 

1.  5  £.  4  s.  10  d.  to  pence.  Ans.  1258  d. 

2.  16  s.  5  d.  3  far.  to  farthings.  Ans.  791  far. 

3.  7s.   1  far.  to  farthings.  Ans.  337  far. 

4.  251  d.  to  pounds.  Ans.  1  £.  11  d. 

5.  100  far.  to  shillings.  Ans.  2  s.  1  d. 

6.  793  s.  to  pounds.  Ans.  39  £.  13  s. 

7.  10  £.  3  d.  to  farthings.  Ans.     ? 

8.  53675  far.  to  pounds.  Ans.     ? 

271. — 111.  FRENCH  MONEY. 

TABLE. 

10  milliemes  (mi-lame)  =  1  centime. 
100  centiemes  (son-teem')  =  1  franc. 

NOTE. — One  franc  is  equal  to  $.186  U.  S.  money. 

MEASURE  OF  TIME. 

272.  The  Units,  or  Measures,  used  in  measuring  time, 
are  the  century  (C.),  the  year  (yr.),  the  month  (mo.),  the 
week  (wk.),  the  day  (d.  or  da.),  the  hour  (hr.  or  h.),  the 
minute  (m.),  and  the  second  (sec.  or  s.X 

N.  I.— 13. 


194 


Intermediate  Arithmetic. 


TABLE. 


60  s.  =  1  m. 
60  m.  =  1  h. 
24  h,  =  1  da. 


7  da.  —  1  wk. 
365  da.  =  1  yr. 


The  Solar  Year  is  exactly  365  da.  5  hr.  48  m.  49.7  sec.,  or  365^ 
days  nearly.  In  four  years  this  fraction  amounts  nearly  to  one 
day.  To  provide  for  this  excess  one  day  is  added  to  the  month 
of  February  every  fourth  year,  which  is  called  Leap  Year  (L.  yr.). 

Every  year,  except  those  ending  with  two  O's,  that  is  exactly 
divisible  by  4  is  a  L.  yr. ;  as  1844,  1856,  1884. 

Every  year  ending  with  two  O's  that  is  exactly  divisible  by  400 
is  a  L.  yr. ;  as  1600,  2000,  2400. 

Every  year  which  is  not  so  divisible  is  a  common  year  ;  as  1847, 
1855,  1900,  1800. 

A  common  year  consists  of  365  days,  a  leap  year  of  366  days, 
and  a  century  of  100  successive  years. 

The  Civil  Year  is  divided  into  twelve  Calendar  months,  thus : 


January  (Jan.)  1st  mo..  .31  da. 
February  (Feb.)  2d  mo..  .28  da. 
March  (Mar.)  3d  mo... 31  da. 
April  (Apr.)  4th  mo.  ..30  da. 
May  (May)  5th  mo..  .31  da. 
June  (June)  6th  mo..  .30  da. 


July  (July)  7th  mo.  31  da. 

August  (Aug.)  8th mo.  31  da. 
September  (Sep.)  9th  mo.  30  da. 
October  (Oct.)  10th  mo.  31  da. 
November  (Nov.)  llth  mo.  30  da. 
December  (Dec.)  12th  mo.  31  da. 


MENTAL  EXERCISES. 


273.  How  many: 

1.  Seconds  in  2  m.?    5m.?    |  m.  ?    J  m.  ?    ^m.? 

2.  Minutes  in  180  s.  ?    90s.?    3  hr.?    J  hr.  ?     £hr.? 

3.  Hours  in  120  m.  ?    600  m.  ?    2  da.  ?    \  da.  ?  \  da.  ? 

4.  Days  in  48  hr.  ?    36  hr.  ?    240  hr.  ?    2  wk.  ?    5  wk.? 

5.  Is  1824  a  common  or  a  leap  year?      1838?     1874? 
1855?    1900?    1700?    1600?    1950?    2200?    2800?  3000? 


Circular  Measure. 


195 


WRITTEN  EXERCISES. 

274.  How  many: 

1.  Days  in  32  common  years?  Ans.  11680  da. 

2.  Days  in  32  leap  years?  Ans.  11712  da. 

3.  Hours  in  5  yr.  120  da.  15  hr.  ?  Ans.  46695  hr. 

4.  Hours  in  10  L.  yr.  106  da,  17  hr.  ?  Ans.  90401  hr. 

5.  Minutes  in  3  wk.  5  da.  10  hr.  12  m.  ?  Ans.     ? 

6.  Weeks,  etc.,  in  583  hr.  ?          Ans.  3  wk.  3  da.  7  hr. 

7.  Years,  etc.,  in  45375204  m.  ? 

Ans.  86  yr.  120  da.  13  hr.  24  m. 

8.  Days,  etc.,  in  1000000  sec.  ?  Ans.    ? 

9.  How  many  days  are  in  the  century  beginning  with 
the  year  1801  and  ending  with  the  year  1900? 

Ans.  36524  da. 

SUGGESTION.— Multiply  365  da.  by  100,  and  to  the  product  add 
as  many  days  as  there  are  L.  yr. 

CIRCULAR  MEASURE. 


275.  The  measures  used  in  meas- 
uring angles  and  the  arcs  of  cir- 
cles are  the  circle  (cir.),  the  degree 
(°),  the  minute  ('),  and  the  sec 

ond  CO- 
TABLE. 

60"=  =  1'.       60'=  1°.       360°=  1  cir. 


EXERCISES. 

276.  How  many  : 

1.  Seconds  in  7°  15'  25"? 

2.  Minutes  in  1  cir.  ? 

3.  Minutes  in  3  cir.  150°  15'? 


CIRCLE. 


Ans.  26125". 
Ans.  21600'. 
Ans.  73815'. 


196  Intermediate  Arithmetic, 


4.  Seconds  in  16°  50"?  Ans.  57650". 

5.  Degrees,  etc.,  in  7453"?  An*.  2°  4'  13". 

6.  Circles,  etc.,  in  584375'?  Ana.  27  cir.  19°  35'. 

7.  Circles,  etc.,  in  73564807"?  Ana.  ? 

8.  Seconds  in  10  cir.  5°  35'  45"?  Ans.  ? 

PAPER  MEASURE. 

277.  The  measures  used  in  measuring  paper  are  the 
bale  (b.),  the  bundle  (bun.),  the  ream  (rm.),  the  quire 
(qr,),  and  the  sheet  (sht). 


TABLE. 


24  sht.  =  l  qr. 
20  qr.  =1  rm. 


2  rm.   =1  bun. 
5  bun.  =  1  b. 


MISCELLANEOUS  TABLE. 


12  units  =  1  dozen. 
12  dozen  =  1  gross. 
20  units  —1  score. 


4  inches     =  1  hand. 
6  feet         =  1  fathom. 
8  furlongs  =  1  mile. 


See,  also,  Art.  113. 

THE  OLD  FRENCH  MEASURE. 

278.  The  old  French  Linear  and  Land  Measure  is 
still  partly  used  in  Louisiana,  and  in  other  French 
settlements  of  the  United  States. 

TABLE. 


12  lines    =  1  inch. 
12  inches  =  1  foot. 


6  feet    =  1  toise. 
32  toises  =  1  arpent. 


1024  sq.  toises  —  1  sq.  arpent. 

The  French  foot  equals  12.79  English  inches. 
The  arpent  is  the  old  French  name  for  acre,  and  is 
equal  to  about  jj-  of  an  English  acre. 


Exercises  in  Reduction.  197 

EXERCISES  IN  REDUCTION. 

279.  Reduce: 

1.  5  R).  8  oz.  11  pwt.  to  grains.          Ans.  32904  gr. 

2.  13  bu.  5  pk.  6  qt.  to  quarts.  Ans.  462  qts. 

3.  2  mi.  45  ch.  to  yards.  Ans.  4510  yd. 

4.  5  ch.  3  yd.  2  ft.  to  inches.  Ans.  4092  in. 

5.  6  yr.  25  da.  6  hr.  to  minutes.        Ans.  3189960  m. 

6.  12  L.  yr.  18  da.  to  hours.  Ans.  105840  hr. 

7.  25  cu.  yd.  15  cu.  ft.  to  cubic  feet.  Ans.  690  cu.  ft. 

8.  16  sq.  rd.  12  sq.  yd.  8  sq.  ft.  to  sq.  ft. 

9.  50  pk.  2  qt.  to  pints.  Ans.  804  pt. 

10.  18  bar.  10  gal.  2  qt.  1  pt.  to  pints.  Ans.  4621  pt. 

11.  13°  25'  to  seconds.  Ans.  48300". 

12.  12  £.  5  s.  11  d.  to  pence.  Ans.  2951  d. 

13.  5  hhd.  15  gal.  1  pt.  to  gills.  Ans.  10564  gi. 

14.  15  A.  3  R.  20  P.  to  poles.  Ans.  2540  P. 

15.  $2,  5  d.  6  c.  to  cents.  Ans.  256c. 

16.  7  hunds.  5  tens,  3  ones  to  ones.  Ans.  753  ones. 

17.  2  lb.  3  3,  4  3,  2  9  to  scruples.  Ans.  662  £. 

18.  $43.75  to  mills.  Ans.  47750  m. 

19.  5  t.  7  cwt.  74  Ib.  to  pounds.  Ans.  10774  Ib. 

20.  7  b.  1  bun.  1  rm.  to  quires.  Ans.  1460  qr. 

Reduce  : 

21.  3779  in.  to  rods.     Ans.  18  rd.  5  yd.  2  ft.  11  in. 

22.  12500  m.  to  days.         Ans.  8  d.  16  h.  2  m. 

23.  4392  P.  to  acres.          Ans.  27  A.  1  R.  32  P. 

24.  24352  far.  to  £.,  etc.        Ans.  25  £.  7  s.  4  d. 

25.  47643  cu.  in.  to  cu.  yds     Ann.  I  cu.  yd.  987  cu.  in. 

26.  1075  gi.  to  gallons.  AIM.  33  gal.  2  qt.  3  gi. 

27.  953  9  to  pounds.  Ans.  3  Ib.  3  £,  5  3,  2  9. 

28.  895  pt.  to  bushels.         Ans.  13  bu.  3  pk.  7  qt.  1  pt. 

29.  8433  qrs.  to  reams.        Ans.  421  rm.  13  qr. 

30.  24563  sq.  in.  to  sq.  yds. 


198  Intermediate  Arithmetic. 

COMPOUND  ADDITION. 

280.  i.  Add  together  5  yd.  2  ft.  9  in. ;  6  yd.  1  ft.  7  in. 
and  4  yd.  2  ft.  4  in.  Ans.  17  yd.  8  in. 

EXPLANATION. — Since  only  like  numbers  can  be    OPERATION. 
added,  we  write  inches  under  inches,  feet  under     yd.    ft.    in. 
feet,  etc.     Adding  the  column  of  inches,  we  get      529 
20  in.,  which,  divided  by  12,  gives  1  ft.  8  in.     Set       Q     ^ 
the  8  in.  under  the  column  of  in.,  and  carry  the 
1  ft.  to  the  column  of  ft. ;  adding  this  column,  we . 


get  6  ft.,  which  equals  2  yd.  and  0  ft. ;   writing  0     17 
under  the  column  of  ft.,  and  carrying  2  to  the  col- 
umn of  yd.,  we  have  17  yds.     Hence,  the 

RULE. — I.  Write  the  numbers  to  be  added  so  that  those  of 
the  same  unit  may  be  in  the  same  column. 

II.  Add  each  column,  beginning  at  the  right,  divide  the 
sum  by  the  number  of  units  of  the  column  added  which 
equals  one  of  the  next  higher,  set  the  remainder  under  that 
column,  and  carry  the  quotient  to  be  added  to  the  next. 

2.  What  is  the  sum  of  9  £.  16  s.  8  d.,  and  10  £.  12  s. 

7  d.  ?  Ans.  20  £.  9  s.  3  d. 

3.  What  is  the  sum  of  7  £.  13  s.  6  d.,  2  £.  17  s.  9  d., 
3  £.  8  s.  3  d.,  9  £.  11  s.  8  d.  ?  Ans.  23  £.  11  s.  2  d. 

4.  What  is  the  sum  of  4  bu.  3  pk.  1  qt.,  7  bu.  2  pk. 
3  qt.,   1  bu.  1  pk.  7  qt.,  and  8  bu.? 

Ans.  21  bu.  3  pk.  3  qt. 

5.  What  is  the  sum  of  5  Ib.  7  oz.  10  dr.,  7  Ib.  11  oz., 

8  dr.,  12  Ib.  5  dr.,  13  Ib.,  3  Ib.  6  oz.  3  dr? 

Ana.  41  Ib.  9  oz.  10  dr. 

COMPOUND  SUBTRACTION. 

281.  1.  From  7  Ib.  5  oz.  9  pwt.  7  gr.  take  3  Ib.  4  oz. 
12  pwt.  4  gr.  Ans.  4  Ib.  17  pwt.  3  gr. 


Compound  Subtraction.  199 

EXPLANATION.  -  Since   only   like   numbers 

can  be  .subtracted,  we  write  gr.  under  gr..  pwt. 

,  I)-.       oz.      pwt.  gr. 

under  pwt..  etc. 

I"  ~  Q          y 

Beginning  at  the  right,  we  subtract  4  gr. 
from   7   gr.  and   get   3   gr.,  which    we   write        °        ^ 
under  the  column  of  gr.  4       0     17     3 

Since    12   pwt.   is   larger  than  9  pwt.,  we 

take  1  oz.  from  the  5  oz.,  leaving  4  oz.,  and  add  it,  or  20  pwt., 
to  9  pwt.,  making  2!)  pwt.  12  pwt.  from  29  pwt.  leave  17  pwt.  ; 
which  we  write  under  the  pwt. 

Since  1  oz.  was  taken  from  5  oz.,  we  subtract  4  oz.  from  4 
oz.  and  get  0  oz.,  which  we  write  under  oz.  3  Ib.  from  7  Ib. 
leaves  4  Ib.,  which  we  write  under  Ib. 

Hence  (see  Art.  87),  the 

RULE. — I.  Write  the  less  number  under  the  greater  so  that 
those  of  the  same  kind  shall  be  in  the  same  column. 

II.  Begin  at  the  right   and   subtract  each  term  from  the 
one  above  it,  if  the  latter  is  the  greater,  and  place  the  dif- 
ference under  the  numbers  subtracted. 

III.  //  any  term  is  greater  than  the  one  above  it,  add  to 
the   one   above  the   number  of  units    of  that  column   which 
equals  one  of  the  next   higher,  from  the  sum   subtract   the 
lower  term,  ivrite  the  remainder  below,  and  carry  one  to  the 
next  term  to  be  subtracted,  and  so  on  with  all  the  columns. 

2.  From  45  A.  2  R.  17  P.  take  19  A.  3  R.  36  P. 

Ans.  25  A.  2  R.  21  P. 

3.  From   65   cu.  yd.  20  cu.  ft.  1252   cu.  in.  take  55 
cu.  yd.  26  cu.  ft.  956  cu.  in. 

An*.  9  cu.  yd.  21  cu.  ft.  296  cu.  in. 

4.  From  85  bu.  2  pk.  take  45  bu.   1  pk.  6  qt. 

Ans.  40  bu.  2  qt. 

5.  From  5  yr.  take  3  yr.  9  mo.         Ans.  1  yr.  3  mo. 

6.  From  12  Ib.  3  3,  1  3,  take  5  Ib.  7  .5,  5  3,  2  B. 

Ans.  6  Ib.  7  3,  3  5,  1 


200  Intermediate  Arithmetic. 

COMPOUND  MULTIPLICATION 

282.  Multiply  3  £  5  s.  9  d.  by  7.  Ans.  23  £  3  d. 

EXPLANATION. — We  write  the  multiplier  under  OPEBATION. 

the  term  on  the  right;   multiply  each  term  as  £>     Si      d> 

in  simple  numbers,  setting  down  and  carrying  as  59 

in  compound  addition.    Thus,  7  X  9  d.  -  -  63  d.,  „ 

which  divided  by  12  gives  5  s.  3  d.     Write  the  3     

below,  and  carry  5s.     7  X  5  s.  -  -  35  s.  and  5  s.  =  23             3 

40  s.,  which  divided  by  20  gives  2  £  0  s.      Write 

the  0  below  and  carry  2  «£.     7  X  3  £  ==  21  £  and  2  £  ==  23  £. 

Hence,  the 

RULE. — Multiply  each  term  of  the  multiplicand,  beginning 
at  the  right,  by  the  multiplier ;  divide  the  product  by  the 
number  of  units  of  the  term  multiplied  which  equals  one  of 
the  next  higher;  set  the  remainder  under  that  term,  and 
carry  the  quotient  to  be  added  to  the  next  product. 

2.  Multiply  22  A.  3  R.  35  P.  by  6. 

Ans.  137  A.  3  R.  10  P. 

3.  Multiply  3  Ib.  4  oz.  0  dr.  2  scr.  by  4. 

Ans.  13  Ib.  4  oz.  2  dr.  2  scr. 

4.  Multiply  13  bu.  2  pk.  1  pt.  by  15. 

Ans.  202  bu.  2  pk.  7  qt.  1  pt. 

5.  Multiply  5  Ib.  3  oz.  13  dr.  by  7. 

Ans.  36  Ib.  10  oz.  11  dr. 

6.  How  many  bushels  in  9  bins,  each  containing  120 
bu.  3  pk.  3  qt.?  An*.  1087  bu.  2  pk.  3  qt. 

COMPOUND  DIVISION. 

283.  1.  Divide  44  bu.  3  pk.  3  qt.  by  7. 

Ans.  6  bu.  1  qt.  5  pk. 


Parallel  Problems.  201 

EXPLANATION. — Dividing  44  bu.  by  7  we  OPERATION. 

get  0  bu.  and  2  bn.  over.    Write  the  0  below,         1)U        pk          t 
reduce  the  2  bu.  to  pk.,  and  to  it  add  3  pk.,      7^44 

which  gives  11  pk.     11  pk.  -f-7  -  =  l  pk.  and  4 

fi.        1  f 

pk.  over.     Write  the  1  below,  reduce  the  4 

pk.  to  qt.,  and  to  it  add  3  qt,  making  35  qt., 

which  divided  by  7  gives  5  qt. ;  write  the  5  below. 

Hence,  the 

RULE. — I.  Place  the  divisor  on  the  left  of  the  dividend, 
and  divide  the  left-hand  term  by  it,  writing  the  quotient 
under  that  term. 

II.  Reduce  the  remainder,  if  any,  to  the  next  lower  unit, 
adding  in  like  terms  of  the  dividend,  if  any,  and  divide  the 
sum  by  the  divisor ;  and  so  on,  for  the  other  terms. 

2.  Divide  24  Ib.  7  oz.  8  pwt.  by  2.  Ans.  12  Ib.  3  oz.  14  pwt. 

3.  Divide  10  hhd.  46  gal.  1  qt.  1  pt.  by  7. 

Ans.  1  hhd.  33  gal.  2  qt.  1  pt. 

4.  Divide  11  £  6  s.  3  d.  by  5.  Ans.  2  £  5  s.  3  d. 

5.  Divide  36  Ib.  10  oz.  9  dr.  by  5.  Ans.    ? 

« 

284.  PARALLEL  PROBLEMS. 

l.™  What  will  4^  quarts  of  plums  cost  at  6  cents  a 
quart?  At  3  cents  a  pint?  At  5  cents  a  pint? 

2.  What  will  5  hogsheads  of  molasses  cost  at  37  cents 
a  gallon?  Ans.  $116.55' 

3.m  If  1  bushel  of  potatoes  cost  80  cents,  what  is  the 
price  of  1  peck? 

4.  If  12  bushels  of  potatoes  cost  $6.48,  what  will  be 
the  value  of  1  peck?  Ans.  13^  c. 

5.™  A  boy  picked  3  quarts  of  cherries,  and  sold  them 
at  the  rate  of  5  cents  a  pint ;  how  much  did  he  receive  ? 

6.  A  farmer    gathered   24  bu.  3  pk.  of  peaches,  and 


202  Intermediate  Arithmetic. 

sold  them  at  the  rate  of  3  cents  a  quart;  how  much  did 

he  receive?  Ans.  $23.76. 

7.m  A  boy  bought  half  a  bushel  of  chestnuts  for  30  c., 

and  sold  them  at  20  c.  a  peck;  how  much  did  he  make? 

8.     A  man  bought  1  seventh  of  a  hogshead  of  sugar 

for  $12.60,  and  retailed  it  at  20  cents  a  pint ;  how  much 

did  he  make?  Ans.  $1.80. 

9.™  How  many  gills  are  there  in  f  of  a  gallon? 

10.  How  many  pints  are  there  in  -}J  of  a  hogshead? 

Ans.  399  pt. 

ll.m  How  many  hours  in  f  of  a  day?  In  T7^  of  a 
day? 

12.     How  many  minutes  in  f  of  a  day?     Ans.  1200  mi. 

13.™  How  far  will  a  man  travel  in  2  hours,  if  he  goes 
5  chains  in  1  minute? 

14.  How  far  will  a  man  travel  in  3  weeks  at  the 
rate  of  5  miles  an  hour?  Ans.  2520  mi. 

15.™  What  will  f  of  a  ream  of  paper  cost  at  20  cents 
a  quire?  At  30  cents  a  quire? 

16.  What  will  |  of  a  bale  of  paper  cost  at  1^  cents 
a  sheet?  Ans.  $28.80. 

I7.m  How  long  will  it  take  a  man  to  travel  100  miles 
if  he  goes  5  miles  an  hour? 

18.  How  long  would  it  take  a  horse,  traveling  at  the 
rate    of  8    miles   an   hour,   to    go   around   the'   world,  a 
distance  of  25000  miles?  Ans.  130  da.  5  hr. 

19.  How  long  would  it  take  a  locomotive  to  go  from 
the  earth  to  the  moon,  a  distance  of  240000  miles,  trav- 
eling at  the  rate  of  25  miles  an  hour  ?     Ans.  1  yr.  35  da. 

20. "i  How  many  times  will  a  wheel  8  feet  in  circum- 
ference turn  over  in  going  120  yards? 

21.  How  often  will  a  wheel  12  feet  in  circumference 
turn  over  in  going  5  miles?  Ans.  2200. 

22. m  If  a  buggy,  whose  wheels  are  12  feet  in  circum- 


Parallel  Problems.  203 

ference,  goes  400  yards  in  10  minutes,  how  often  do  the 
wheels  turn  over  in  1  minute? 

23.  If  a  locomotive,  whose  wheels  are  15  feet  in  cir- 
cumference, runs  at  the  rate  of  45  miles  an  hour,  at 
what  rate  do  the  wheels  revolve  per  minute?  Am.  264. 

24. m  What  will  it  cost  to  build  a  fence  2  miles  long 
at  SI  a  chain  ? 

25.  What  will  36  miles  of  telegraph  wire  cost  at  75 
cents  a  rod  ?  Ans.  $4320. 

26.f"  If  a  lad  makes  5  steps  in  walking  a  rod,  how 
many  steps  will  he  make  in  going  3  chains? 

27.  If  20  rails  are  required  to  build  a  fence  1  rod 
long,  how  many  rails  will  it  take  to  inclose  a  field  ^  of 
a  mile  long  and  ^  of  a  mile  wide?  Ans.  2400  rails. 

285.  QUESTIONS  FOR  REVIEW. 

What  is  a:  1.  Simple  number?  2.  Compound  number?  State 
what  each  of  the  following  measures  are  used  for,  name  the  meas- 
ures, and  repeat  the  Table :  1.  Linear  Measure;  2.  Square  Meas- 
ure ;  3.  Solid  or  Cubic  Measure ;  4.  Liquid  Measure  ;  5.  Dry  Meas- 
ure ;  6.  Troy  Weight ;  7.  A  voirdupois  Weight ;  8.  Apothecaries' 
Weight;  9.  English  Money;  10.  Measure  of  Time;  11.  Circular 
Measure  ;  12.  Paper  Measure. 

What  is  the  rule  for:  1.  Reduction  Descending?  2.  Reduction 
Ascending  ? 

How  many:  1.  Cubic  inches  in  a  gallon?  2.  Cubic  inches  in  a 
bushel  ?  3.  Dollars  in  1  £  ?  4.  Cents  in  1  franc  ?  5.  Days  in  a 
leap-year  ?  6.  Acres  in  an  arpent  ? 

Which  are  the:  1.  Leap-years?    2.  Common  years? 

Name  the  number  of  days  in  each  month. 


IMPORTANT  APPLICATIONS. 


286.  CASE  I. — To  find  the  time  between  two  dates. 

1.  What  length  of  time  elapsed  from  July  10,  1843, 
to  Jan.  4,  1845  ? 

EXPLANATION. — "Write  the  latter  or  greater 
date  for  the  minuend,  and  the  earlier  for  the  OPERATION. 

subtrahend,  giving  the  month  its  number  in-        yr  mo        da 

stead  of  the  name.     Thus,  since  Jan.  is  the      1845          1          4 
1st  month,  we  write  1  under  mo.  and  under      1843          7        JQ 

it  write  7,  as  July  is  the  7th  month.     Now   

subtract  as  in  compound  subtraction,  allow-  5 

ing  12  mo.  to  the  yr.  and  30  da.  to  the  mo. 

Find  the  time  from  : 

2.  May  12,  1848,  to  June  1,  1860.  Ans.  12  yr.  19  da. 

3.  June  1,  1861,  to  Oct.  20,  1872. 

Ans.  11  yr.  4  mo.   19  da. 

4.  June  15,  1846,  to  Jan.  10,  1848. 

Ans.  1  yr.  6  mo.  25  da. 

5.  April  20,  1868,  to  Aug.  1,  1869. 

Ans.  1  yr.  3  mo.  11  da. 

6.  Henry  was  born  March  9,  1868,  and  Harry  Sept. 
15,  1875 ;  how  much  older  is  Henry  than  Harry  ? 

Ans.  7  yr.  6  mo.  6  da. 

7.  A  note   dated  July    15,   1879,   was   paid  May  21, 
1882;  how  long  did  it  run?         Ans.  2  yr.  10  mo.  6  da. 

To  what  age  did  the  following  live : 

8.  Washington,   born    Feb.   22,   1732;    died    Dec.    14, 
1799? 

(201) 


Important  Applications. 


205 


9.  Jefferson,  born   April  2,   1743;  died  July  4,  1S20? 

10.  Lincoln,  born  Feb.  12,  1S09;  died  April' 15,  18G5? 

11.  Calhoun,   born    March   18,   17*2;   died   March   31, 
1850? 

12.  Webster,  born  Jan.  18,  1782;  died  Oct.  24,  1852? 

13.  Clay,  born  April  12,  1777;  died  June  29,  1852? 

287.  CASE  II. — To  find  the  area  of  rectangular  sur- 
faces. 

A  rectangular  surface  is  a  surface  in  the  shape  of  a 
sheet  of  paper,  or  of  an  ordinary  square  cornered  gar- 
den of  four  sides. 

The  figure  represents  a 
rectangular  surface  5  in. 
long  and  3  in.  wide,  and 
evidently  contains  3x5= 
15  square  inches.  15  sq. 
in.  is  called  its  area. 

Hence,  the 

RULE. — Multiply  the  length  by  the  width,  expressed  in 
like  units. 

1.  What  is  the  area  of  a  floor  18  ft.  long  and  12  ft. 
wide?  Ans.  216  sq.  ft.,  or  24  sq.  yd. 

2.  What   is  the  area  of  a  rectangular  garden   88  yd. 
long  and  55  yd.  wide?  Ans.  4840  sq.  yd.,  or  1  A. 

3.  What  is  the  area  of  a  rectangular  field  35  ch.  long 
and  24  ch.  wide?  Ans.  840  sq.  ch.,  or  84  A. 

4.  How  many  acres  in  a  meadow  17  ch.  long  and  12 
ch.  wide?  Ans.  20.4  A. 

5.  How  many  acres  in  a  field  60  rd.  long  and  44  rd. 
wide?  Ans.  16£  A. 

6.  How  many  yards  of  carpeting  3  ft.  wide  will  it  take 
to  cover  a  floor  20  ft.  long  and  15  ft.  wide  ? 


206 


In ter  mediate  Arithmetic. 


We  divide  the  number  of  sq.  ft.,  300,  OPERATION. 

by  the  width  of  the  carpet,  3  ft.,  which        20  X  15  =  300 
gives  the  length  of  the  carpet,  100  ft.,        300  -=-3  —  100 

and  this  divided  by  3  gives  33£  yd.  1QQ  ft.  -r-  3  =  33J  yd. 

7.  How   many   yards  of  carpeting  4   ft.  wide   will   it 
take  to  cover  a  floor  22  ft.  long  and  18  ft.  wide? 

Ans.  33  yd. 

288,  CASE  III.— To  find  the  volume  of  a  rectangular 

- 

solid. 

A  rectangular  solid  is  a  solid  in  the  shape  of  an  or- 
dinary goods-box,  or  of  a  room. 

The  figure  represents  a  rectan- 
gular solid  5  ft.  long,  4  ft.  high,  3 
ft.  wide,  and  evidently  contains 
5  x  4  X  3  =  60  cu.  ft.  60  cu.  ft.  is 
called  its  volume.  Hence, 

RULE. — Multiply  the  length, 
width,  and  height  together. 

NOTE. — The  dimensions  must 
be  expressed  in  terms  of  the  same 
measure. 

1.  What  is  the  volume  of  a  rectangular  box  6  ft.  long, 
3  ft.  wide,  and  5  ft.  high  ?  Ans.  90  cu.  ft. 

2.  What  is  the  volume  of  a  room  20   ft.  long,  18  ft. 
wide,  and  12  ft.  high?      Ans.  4320  cu.  ft.,  or  160  cu.  yd. 

3.  How  many  cubic  feet  in  a  marble  slab  60  in.  long, 
24  in.  wide,  and  6  in.  thick  ?  Ans.  5  cu.  ft. 

4.  How  many  cubic  yards  of  coal  in  a  pile  18  ft.  long. 
10  ft.  wide,  and  7  ft.  high  ?  Atis.  46|  cu.  yd. 

5.  How  many  cubic  feet  of  corn  in  a  crib  20  ft.  long, 
and  12  ft.  wide,  the  corn  being  5^  feet  deep  in  the  crib  ? 

Ans.  1320  cu.  ft, 


Important  Applications.  207 

6.  How  many  cords  of  wood  in  a  pile  20  ft,  long,  12 
ft.  wide,  and  6  ft.  high?  Ans.  Hi  cords. 


OPERATION.  *  OX£  2X6  ^ 


How  many  cords  in  a  pile  of  wood  : 

7.  24  ft.  long,  8  ft.  wide,  and  4  ft.  high  ?      Ans.  6  c. 

8.  18  ft.  long,  6  ft.  wide,  and  8  ft.  high  ?        Ans.  6f  c. 

289.  CASE   IV.--  To  find  the  number  of  gallons  in  a 
rectangular  box  or  tank. 

1.   How    many   gallons    of  water   will   a   tank   hold 
which  is  6  ft.  long,  5  ft.  wide,  and  8  ft.  high? 

OPERATION.  6X5X8X7^  =  1800  gal.    Ans. 

EXPLANATION.  —  Since  there  are  231  cu.  in.  in  one  gallon,  and 
1728  cu.  in.  in  one  cubic  foot,  1  cu.  ft.=-y^2T8-  gal.  or  about  7%  gal. 

Hence,  the 

RULE.  —  Multiply  the  number  of  cu.  ft.  by  7J. 

2.  How  man}'-    gallons  of  molasses   will  a   box    hold 
which  is  4  ft.   long,  2  ft.  wide,  and  9  in.  (f  ft.)  high? 

Ans.  45  gal. 

3.  A  box   is  3  ft.  long,  2  ft.  wide,  8  in.  (f  ft.)   high, 
and  is  full  of  honey  ;  how  much  honey  is  in  the  box  ? 

Ans.  30  gal. 

What  is  the  capacity  of  a  tank  which  is  : 

4.  6  ft.  long,  4  ft.  wide,  and  10  ft.  high  ? 

Ans.  1800  gal. 

5.  4  ft.  long,  4  ft.  wide,  and  9  ft.  high? 

Ans.  1080  gal. 

6.  5  ft.  square  at  the  bottom  and  7  ft.  high  ? 

Ans.  1312J  gal. 


208  Intermediate  Arithmetic. 

7.  3  ft.  square  at  the  bottom  and  6  ft.  high? 

Ans.  405  gal. 
How  much  milk  will  a  box  hold  which  is : 

8.  1|  ft.  long,  1-J-  ft.  wide,  and  J  ft.  high?  Ans.  7-J-  gal. 

9.  If  ft.  long,  1£  ft.  wide,  and  6  in.  high? 

Ans.  7\  gal. 
10.  9  in.  long,  8  in.  wide,  and  4  in.  deep? 

Ans.  1J  gal. 

290.  CASE  V. — To  find  the  number  of  bushels  in  a 
rectangular  box,  bin,  or  granary. 

1.  How  many  bushels  of  wheat  will  a  box  hold  which 
is  5  ft.  long,  4  feet  wide,  and  3  ft.  deep? 

OPERATION.          5X4X3X.8  =  48.0  =  48  bu. 

EXPLANATION. — Since  there  are  2150.4  cu.  in.  in  one  bushel, 
and  1728  cu.  in.  in  one  cubic  foot,  then  1  cu.  ft.  =  2T^?bu.— 
about  .8  bu. 

Hence,  the 

RULE. — Multiply  the  number  of  cu.  ft.  by  .8. 

How  many  bushels  will  a  bin  hold  : 

2.  Which  is  6  ft.  long,  5  ft.  wide,  and   2|   ft.  high? 

Ans.  60  bu. 

3.  Which  is  6  ft.  long,  4|  ft.  wide,  and  2|  ft  deep? 

Ans.  57f  bu. 

4.  Which  is  5-J  ft.  long,  3f  ft.  wide,  and  2£  ft.  deep? 

Ans.  40  bu. 

5.  Which  is  7J-  ft.  long,  2-J  feet  wide,  and  3£  ft.  deep? 

Ans.  53^-  bu. 

291.  CASE  VI. — To  find  the  number  of  board  feet  in 
boards  or  planks. 

A  Board  Foot  is  used  in  measuring  boards,  planks, 
and  sawed  timber  generally.  It  is  1  foot  long,  1  foot 
wide,  and  1  inch  thick,  and  contains  144  cu.  in. 


Important  Applications.  209 

l.  How  many  board  feet  in  a  plank  13  ft.  long,  18 
in.  wide,  and  2  in.  thick?  Am.  39  B.  ft. 


OPERATION. 

13  X  18  X  2 


12 
Hence,  the 


=  39. 


.--  Multiply   the   length   in  feet   by  the  width  and 
thickness  expressed  in  inches,  and  divide  the  product  by  12. 

NOTE.  —  If  a  board  or  plank  is  less  than  1  in.  thick,  it  is  dis- 
regarded; that  is,  the  calculation  is  made  as  if  it  were  1  in. 
thick. 

2.  What   is   the   number  of  feet   in   a  board   15   ft. 
long,  9  in.  wide,  and  ^  in.  thick.  Ans.  11  J  B.  ft. 

3.  How  many  feet  in  12  planks,  each   7  ft.  long,  8 
in.  wide,  and  1|-  in.  thick?  Ans.  84  B.  ft. 

4.  How  many  feet  in  9  boards,  each  11  ft.  long,  6 
in.  wide,  and  f  in.  thick?  Ans.  49  \  B.  ft. 

5.  How  many  feet  in  36  plank,  each  10  ft.  long,  11 
in.  wide,  and  1^  in.  thick?  Ans.  440  B.  ft. 

292.  PARALLEL  PROBLEMS. 

l.m  Albert  was  born  Feb.  17,  1868,  and  Robert  Feb. 
17,  1875;  how  much  older  is  Albert  than  Robert? 

2.  The  Brooklyn  Suspension  Bridge  was  commenced 
Jan.  3,  1870,  and  was  opened  for  travel  July  4,  1882  ; 
how  long  was  it  in  building? 

Ans.  12  yr.  6  m.  1  da. 

3.m  How  many  acres  in  a  field  8  ch.  long  and  5  ch. 
wide  ? 

4.  How  many  acres  in  a  field  45  rd.  long,  and  38 
rd.  wide?  Ans.  10  A.  2  R.  30  P. 

N.  L—  14. 


210  Intermediate  Arithmetic. 

5.m  If  carpeting  is  4  ft.  wide,  how  many  yards  will 
it  take  to  cover  a  stage  12  ft.  wide,  and  5J  ft.  deep? 

6.  If  carpeting  is  5  ft.  6  in.  wide,  how  many  yards 
will  be  required  to  cover  a  floor  16^-  ft.  long,  and  15^ 
ft.  wide?  Ans.  15  J  yd. 

7.m  How  many  marble  slabs,  3  ft.  long  and  2  ft. 
wide,  are  necessary  to  pave  a  walk  150  ft.  long,  and  4 
ft.  wide? 

8.  How  many  bricks,  8  in.  long  and  4  in.  wide, 
are  required  to  construct  a  pavement  285  ft.  long,  and 

7  ft.  4  in  wide?  Ans.  9405. 

9.™  What  will  it  cost  to  cut  a  ditch  60  ft.  long,  3 
ft.  wide,  and  2  ft.  deep,  at  $J  per  cu.  yd? 

10.  A  street  is  600  ft.  long,  and  55  ft.  wide;  \vhat 
will  be  the  cost  of  elevating  it  3  ft.,  at  60c.  per  cu. 
yd.?  Ans.  $2200. 

11. "i  A  tin  box  3  ft.  long,  2  ft.  wide,  and  1  ft,  high, 
is  full  of  honey;  what  is  the  honey  worth  at  50c.  per 
gallon  ? 

12.  A  cistern  6  ft.  square  at  the  bottom,  and  10  ft. 
high,  is  2  thirds  full  of  water ;  how  long  will  it  supply 
a  family  that  uses  45  gallons  of  water  per  day  ? 

13.m  A  man  sells  corn  at  $^  per  bushel ;  what  should 
be  his  price  for  a  box  of  corn  5  ft  long,  3  ft.  wide,  and 
2  ft.  deep? 

14.  Corn  weighs  56  pounds  per  bushel;  what  is  the 
weight  of  the  corn  in  a  granary  9  ft.  long,  5  ft.  wide, 
and  6  ft.  high,  the  granary  being  3  fourths  full  ? 

15.™  If  lumber  is  worth  $1-J-  per  hundred  board  feet, 
what  is  the  value  of  20  planks,  each  10  ft.  long,  6  in. 
wide,  and  1  in.  thick? 

16.  If  lumber  is  worth  $18.50  per  thousand  board 
feet,  what  is  the  valne  of  36  plank,  each  16  ft.  long, 

8  in.  wide,  1  in.  thick?  Ans.  $7.104. 


p 


ERCENTAGE 


293.  The  term  per  cent  means  by  the  hundred. 
Thus  :  3  per  cent  means  3  hundredths,  or  yf-Q,  or  .03; 

5  per  cent  means  5  hundredths,  or  yf-g-,  or  .05 ;   13  per 
cent  means  13  hundredths,  or  j1^,  or  .13. 

294.  The  process  of  calculating  by  hundredths  is  called 


Percentage. 


INDUCTIVE  EXERCISES. 


295.    1.   A  man  had    100  sheep,  but  lost  7  of  them  ; 
what  part  of  his  sheep  did  he  lose? 

Ans.  yJo-,  or  7  per  cent. 

2.  A   man  had  50  sheep,  but  lost  3  of  them ;    what 
part  of  his  sheep  did  he  lose? 

Ans.  •fo  =  jfaj  or  6  per  cent. 

3.  A  man   had   25   hogs,  but   lost   7   of  them  ;    what 
part  of  his  hogs  did  he  lose? 

Ans.  77T  =  T2A  =  28  per  cent. 

±1   O  i    v   v  -L 

4.  A  man  had  $200,  and  gained  $13  more ;  what  part 
of  his  money  was  his  gain  ? 

Ans.  -£fo  =  -f^  -  -  6^  per  cent. 

5.  What  per  cent  of  a  number  is  ^  of  it? 

Ans.  ^  =  y^  =  =  50  per  cent. 

6.  What  per  cent  of  a  number  is : 
-i  of  it?       Ans.  331 


i  of  it  ?       Ans.  25. 
i  of  it  ?       Ans.  20. 


1  of  it? 
|  of  it? 
i  of  it  ? 

i   of  it  ? 
TV  of  it  ? 
A  of  it? 

2*0  of  it? 
A  of  it? 
^r  of  it? 

(211) 


212  Intermediate  Arithmetic. 

What  fractional  part  of  a  number  is  20  per  cent  of 
it?  A™     20 

7.  What  fractional  part  of  a  number  is : 

25  per  cent  of  it?  Ans.  ^. 
50  per  cent  of  it?  Ans.  \. 
75  per  cent  of  it?  Ans.  f. 


£  per  cent  of  it  ? 
100    per  cent  of  it? 
12^  per  cent  of  it? 


DEFINITIONS. 

296,  Rate  is  a  given  allowance. 

297,  When   rate   is   expressed   by    the    hundred,   the 
number  of  hundredths  taken  or  allowed  is  the  Rate  Per 
Cent. 

298,  Percentage  is  the  result  obtained  by  taking  a  cer- 
tain per  cent  of  any  given  number. 

299.  The  Base  is  the  number  on  which  the  percentage 
is  computed. 

Per  cent  is  indicated  by  %.    Thus,  6%  is  read :  6  per  cent. 

300.  CASE  I.— The  Base  and  Rate  %  given  to  find  the 
Percentage. 

EXERCISES. 

301.  1.  What  is  6%  of  150  sheep? 

ANALYSIS.— 6%  of  150  sheep  is  Tf  ^  of  150  sheep,  which  is  9  sheep. 

Hence,  the 

RULE. — Multiply  the  base  by  the  rate  expressed  decimally. 

How  much  is : 


2.  2      of450bu.  ?  Ans.  9bu. 


3.  15      of  240  c.?    Am.  36  c. 


4.  Sty  of$120?^7is.  810.20. 


5.  20%  of  $185.70?    Ans.  ? 

6.  50%  of  $124.37?    Ans.  ? 

7.  331^  of  183  lb.?     Ans.  ? 


Percenfaf/e.  213 

How  many  dollars  are  made  by  selling : 

8.  A  cow  which  cost  $18,  so  as  to  gain  20%  ? 

Ans.  $3.60. 

9.  A  horse  which  cost  $135,  so  as  to  gain  15^  ? 

Ans.  $20.25. 

10.  A  wagon  which  cost  $84,  at  an  advance  of  12-J-^  ? 

Ans.  310J. 
How  many  dollars  are  lost  by  selling  : 

11.  A  lot  which  cost  $2345,  so  as  to  lose  9%  ? 

Ans.  $211.05. 

12.  A  carriage  which  cost  $245,  so  as  to  lose  8%  ? 

Ans.  $19.60. 

13.  A  saddle  which  cost  $14,  at  a  discount  of  1\%  ? 

Ans.  ? 

302.  CASE  II. --The  Base  and  Percentage  given  to  find 
the  Rate  %. 

EXERCISES. 

303.  1.  What  per  cent  of  16  is  4? 
ANALYSIS. — 4  is  T\  of  16.    -&  =  \  =  -fifa  =  25  per  cent. 

Hence,  the 

RULE. — Multiply  the  number  which  is  the  percentage  by 
100,  and  divide  the  product  by  the  base. 

What  per  cent  of: 

2.  500  is  40?        Ans. 

3.  825  is  50?        Ans. 

4.  $637  is  $38.22?  ,4ns.  6%. 

5.  $300 is $37.50? Ans.12^%. 


6. 

7.  £40 is £2  16s.?  Ans. 

8.  50  gal.  is  2  gal.  3  qt.lAns.  ? 

9.  20yd.  is  5  ft.  10  in.?  Ans.? 


10.  A  man  bought  a  cow  for  $40,  and  sold  her  so  as 
to  gain  $10;  what  per  cent  did  he  make?       Ans.  15%. 

11.  A  man  bought  a  cow  for  $50,  and    sold   her  at  a 
loss  of  $10;  what  per  cent  did  he  lose?  Ans.  20%. 


214  Intermediate  Arithmetic. 


COMMISSION. 

304.  Commission  is  a  sum  of  money  paid  to  an  agt 
for  buying  or  selling  goods  or  other  property.      It  is  a 
percentage  of  the  amount  invested  or  collected. 

MENTAL  EXERCISES. 

305.  How   much   does   an   agent  get   for  his   services 
who  collects : 

1.  $500,  and  charges  2  per  cent?  Ans.  $10. 

2.  $650,  and  charges  4  per  cent?  Ans.    ? 

3.  $200,  and  charges  2^  per  cent?  Ans.   ? 

4.  $325,  and  charges  8  per  cent?  Ans.    ? 

How  much  does  an  agent  receive  who  sells  goods  to 
the  amount  of: 


5.  $450,  at  2%  commission?  Ans. 

6.  $125,  at  8%  commission?  Ans.   ? 

7.  $50,  at  3J%  commission?  Ans.   ? 

WRITTEN  EXERCISES. 

306.  1.  A  commission  merchant  sold  wheat  for  $1900; 
what  was  his  commission  at  2f  per  cent?      Ans.  $52.25. 

2.  A  commission  merchant  invested  $1250  for  another 
party ;  what  was  his  commision  at  If  per  cent  ? 

Ans.  $20. 

What   does  a  commission   merchant   receive   for   his 
services  who  sells : 

3.  5  horses  at  $150  each,  at  2^  per  cent  commission? 

Ans.  $18.75. 


Profit  and  Loss.  215 

4.  12   bales  of  cotton,   averaging   450   pounds    to  the 
bale,  at  8  cents  a  pound,  commission  being  If  per  cent  ? 

Ans.  $7.20. 

5.  0  hlids.   of  sugar,   averaging    1450    pounds   to   the 
lihd.,   at    5    cents   a   pound,   commission    being    If   per 
cent?  An*.  $7.61}, 

6.  120  bu.  apples  @  $1.50,  020  Ib.  butter  @  $.25,  1250 
Ib.  bacon  @  14  c.,  commission  being  \\%  ?     Ans.  $8.92^-. 


PROFIT  AND  Loss. 

307.  Profit  and  Loss  denote  the  gain  or  loss  in  busi- 
ness transactions.     They  are  computed  by  percentage. 

308.  The  cost  is  the  base;  the  per  cent  of  gain  or  loss, 
the  rate  ;  the  gain  or  loss,  the  percentage.     In  case  of  a 
gain,  the   selling  price  is  the  amount,  and  in  case  of  a 
loss,  the  selling  price  is  the  difference. 

309.  PARALLEL  PROBLEMS. 

l  .  A  man  bought  a  gun  for  $8  and  sold  it   for  $10  ; 
what  per  cent  did  he  make?  Ans. 


NOTE.—  The  gain  was  $10  —  $8  =$2,  and  the  cost  $8,  and  by 
Art.  303,  $2  are  25$,  of  $8. 

What  per  cent  does  a  man  make  who  buys  : 

2.m  A  saddle  for  $10,  and  sells  it  for  $12?   Ans. 
3.     A  book  for  $42,  and  sells  it  for  $63?      Ans. 
4.m  A  table  for  $3,  and  sells  it  for  $6?          Ans.   ? 
5.     A  cow  for  $120,  and  sells  her  for  $220?  Ans.  8 
6.m  A  horse  for  $60,  and  sells  him  for  $90?  Ana.  ? 


216  Intermediate  Arithmetic. 

7.  A  man  bought  a  coat  for  $10,  and  sold  it  for 
what  per  cent  did  he  lose?  Ans.  20^. 

NOTE.— The  loss  was  $10—  $8  ==$2,  and  the  cost  $10,  and  by 
Art.  303,  $2  are  20%  of  $10. 

What  per  cent  does  a  man  lose  who  buys : 

8.m  A  saddle  for  $12,  and  sells  it  for  $10?  Ans. 
9.     A  secretary  for  $62,  and  sells  it  for  $40? 


10.™  A  table  for  $6,  and  sells  it  for  $3?          Ans.  50%. 
11.     A  cow  for  $25,  and  sells  her  for  $23?     Ans.  8  ft. 
12.™  A  horse  for  $80,  and  sells  him  for  $60?         Ans.  ? 

13.  A  horse  for  $80.50,  and  sells  him  for  $62?     Ans.  ? 

14.  A  farm  for  $320,  and  sells  it  for  $280?  Ans.  Vl 


What  will  a  merchant  receive   for  an  article  which  : 

15.  m  Cost  $10,  if  sold  so  as  to  gain  20%  ?          Ans.  $12. 

16.  Cost  $122,  if  sold  at  a  profit  of  25%  ?  Ans.  $152.50. 
17.™  Cost  $24,  if  sold  at  a  gain  of  5%  ?  Ans.  ? 
18.     Cost  $282,  if  sold  at  an  advance  of  15%  ?  Ans.  ? 
19.™  Cost  $8,  if  sold  so  as  to  lose  50%  ?  Ans.  $4. 

20.  Cost  $53,  if  sold  at  a  loss  of  22%  ?      Ans.  $41.34. 

21.  m  Cost  $25,  if  sold  at  a  loss  of  12  Jg  ?  Ans.  ? 

22.  Cost  $182,  if  sold  at  a  discount  of  Yl\     ?  Ans.  ? 


310.  QUESTIONS  FOR  REVIEW. 

What  is:  1.  Rate?  2.  Rate  per  cent?  3.  Percentage?  4.  The 
Base? 

How  is  per  cent  indicated  ? 

How  do  we  find :  1.  The  percentage,  when  the  base  and  rate  <J0 
are  given?  2.  The  rate  </c,  when  the  base  and  percentage  are 
given  ? 

What  is:  1.  Commission?    2.  Profit  and  Loss? 


INTEREST, 


INDUCTIVE  EXERCISES. 

311.  Do  men  pay  for  the  use  of  horses  borrowed  from 
a  livery  stable  ?  What  is  tjie  money  so  paid  called  ? 
Ans.  Hire.  How  is  hire  estimated?  Ans.  On  1  horse 
for  1  day.  If  1  horse  for  1  day  cost  $2,  what  do  we 
call  the  $2?  Ans.  The  rate  of  hire.  If  the  rate  of  hire 
is  $2,  what  will  be  the  hire  of  2  horses  for  3  days? 
4  horses  for  5  days  ?  6  horses  for  3  J  days  ? 

Do  men  pay  for  the  use  of  borrowed  land?  What  is 
the  money  so  paid  called?  Ans.  Rent,  How  is  rent 
estimated?  Ans.  On  1  acre  for  1  year.  If  1  acre  for  1 
year  cost  $2,  what  do  we  call  the  $2?  Ans.  The  rate 
of  rent.  If  the  rate  of  rent  is  $4,  what  will  be  the  rent 
of  2  acres  for  1  year?  40  acres  for  1  year?  40  acres 
for  2  years?  80  acres  for  1-J-  years?  50  acres  for  2 
years  6  months,  (2-J  years)  ? 

Do  men  pay  for  the  use  of  borrowed  money  ?  What 
is  the  money  so  paid  called  ?  Ans.  Interest.  How  is 
interest  estimated?  Ans.  On  1  dollar  for  1  year?  If 
$1  for  1  year  cost  6  c.,  what  do  we  call  the  6  c.  ? 
Ans.  The  rate  of  interest.  If  the  rate  of  interest  is  6  c., 
what  will  be  the  interest  of  $20  for  1  year  ?  $20  for  2 
years?  $40  for  2  years? 

What  part  of  $1  is  6  c.  ?  Ans.  .06,  or  6  per  cent. 
Instead  of  saying  "at  the  rate  of  6  c.  on  81,"  would  it 
be  the  same  to  say :  "  at  the  rate  of  6  per  cent  ? " 
What  does  "  at  the  rate  of  8  per  cent ':  mean  ?  Ans. 
"  At  the  rate  of  8  c.  on  $1." 

(217) 


218  Intermediate  Arithmetic. 

DEFIN  ITIONS. 

312.  Interest  is  the  sum  paid  for  the  use  of  mone)T. 

313.  Principal  is  the  money  for  the  use  of  which  in- 
terest is  paid. 

314.  Amount  is  the  sum  of  principal  and  interest. 

Rate  per  cent  is  the  number  of  cents  allowed  for  the 
use  of  $1  for  1  year. 

315.  CASE  I.— When  the  time  is  expressed  in  years. 

1.  What  is  the  interest  of  $40  for  2  years  at  6  per 
cent  ? 

ANALYSIS.— Since  the  rate  is  6%,  the  interest  of  $1  for  1  year  is 
6  c.  Hence,  the  interest  of  $40  for  1  year  will  be  40  times  6  c.  = 
240  c.,  or  $2.40,  and  for  2  years  2  times  $2.40  =  $4.80. 

MENTAL  EXERCISES. 

316.  What  is  the  interest,  at  6  per  cent,  of: 


2.  $3  for  1  yr.  ?  Ans.  18  c. 

3.  $3  for  2  yr.  ?  Ans.  36  c. 

4.  $5  for  2  yr.  ?  Ans  ? 

5.  $10  for  3  yr?.  Ans.  $1.80. 


6.     $4  for  2i    r.  ?  Ans.  60  c. 


7.  $5  for  3Jyr.?  Ans.  $1. 

8.  $10  for  11  yr.  ?  Ans.  ? 

9.  $15  for  \  yr.  ?  Ans.  ? 


Hence,  the 

RULE. — Multiply  the  principal  by  the  rate  per  cent,  and 
the  product  will  be  the  interest  for  1  year;  then  multiply  this 
product  by  the  number  of  years. 

To  find  the  Amount,  add  the  Principal  to  the  Interest. 


Interest. 


219 


WRITTEN  EXERCISES. 

317.  What  is  the  interest  of: 

1.  $120  for  3  yr.  at  6%  ?    At  8%  ?  An*.  $21.60  ;  $28.80. 

2.  $120.40  for  4  yr.  at  5^  ?    At  1%  ? 

-4ns.  $24.08 ;  $33.712. 

3.  $325  for  2  yr.  at  9^  ?     At  3%  ?   Ana.  $58.50 ;  $19.50. 

4.  $234.26  for  5  yr.  at  6%  ?     At  10%  ? 

Ans.  $70.278;  $117.13. 

5.  $180.04  for  2i-  yr.  at  8%  ?     At  7%  ? 

-4n«.  $36.008 ;  $31.507. 

6.  $144  for  H  yr.  at  6%  ?     At  8%  ?         4ws.   ?   ;     ? 

What  is  the  amount  of: 


7.  $49  for  2  yr.  at  5%  ?     At  6%  lAns.  53.90;  $54.88. 

8.  $325  for  3  yr.  at  4%  ?    At  b%  ?  Ans.  $364  ;  $373.75. 

9.  $63.75  for  3J  yr.  at  8%  ?      At  10^  Ana.    ?    ;     ? 
10.  What  is  the  interest  of  $275   at   6%    from  June 

12,  1881,  to  June  12,  1884?     At  $%  ?    Ana.  $49.50;  $66. 

318.  CASE  II.  -  -When  the  time  is  expressed  in  years 
and  months. 

In  this  case  we  reduce  the  time  to  years. 

What  part  of  1  year  is : 

1.  7  months? 

2.  5  months? 

3.  11  months? 

4.  1  month? 

5.  6  months? 

6.  8  months? 


Ans.  -j7^-. 

7. 

^4?is.  T5Y. 

8. 

^4?is.     ? 

9. 

.4ns.     ? 

10. 

Ans.    |-. 

11. 

-4ns.    -|. 

12. 

10  months  ?  Ana.  •}-£  =  ? 

2  months  ?        Ans.  =  J-. 

3  months?         Ans.     J. 
9  months  ?         Ans.  =  ? 

4  months?         Ans. 


o 


How  many  years  in  : 

13.4  yr.  and  3  mo.  ?  Ans.  4J. 
14.  3  yr.  and  8  mo.  1  Ana.  3J. 


15. 
16. 


12  months  ?       Ans.  = 

5  yr.  and  6  mo.  ?  Ana. 
2  yr.  and  11  mo.  ?  Ans. 


220 


Intermediate  Arithmetic. 


WRITTEN  EXERCISES. 

319.  1.  What  is  the  interest 
of  $130.16  at  6%  for  2  yr. 


9  mo.? 

EXPLANATION. —  We  multiply 
the  interest  for  1  year  by  the 
time  (2  yr.  9  mo.)  reduced  to  yr. 
(2f). 

Hence,  the 

RULE. — Reduce  the  time  to 
years  and  proceed  as  in  Case 
I. 


OPERATION. 

130.16     Principal. 
.06     Rate 


7.8096     Int.  for  1  year. 
2|  No.  yr. 


4)23.4288     Product  by  3. 

5.8572     Quotient  by  4. 
15.6192     Product  by  2. 


21.4764     Interest. 


What  is  the  interest  and  amount  of: 

2.  $936  for  2  yr.  5  mo.  at  1%  ? 

Ans.  Int.  $158.34;  Am't  $1094.34. 

3.  $1248  for  3  yr.  8  mo.  at  $%  ? 

Ans.  Int.    ?  Am't  $1476.80. 

4.  $672.84  for  4  yr.  7  mo.  at  $%  ? 

Ans.  Int.  $246.708;  Am't    ? 

5.  $576.48  for  3  yr.  5  mo.  at  6%  ? 

Ans.  Int.  ?   Am't  $694.6584. 

6.  $120.60  for  1  yr.  1  mo.  at  9%  ?    Ans.  Int.  ?  Am't? 

7.  $864.18  for  3  yr.  10  mo.  at  8%  ? 

Ans.  Int.    ?  Am't  $1129.1952. 

8.  $960.48  for  9  mo.  at  b%  ? 

Ans.  Int.  $36.018;    Am't  $996.498. 

9.  What  is  the  interest  of  $320  from  May  12,  1878, 


to  July  12,  1881,  at 


Ans.  $60.80. 


10.  What  is  the  amount  of  $20.15  from  Dec.  17,  1880, 
to  March  17,  1885,  at  8%  ?  Ans.  $27.001. 

11.  A  man  borrowed  $180.60  June  7,  1881,  and  settled 
the  account  Oct.  7,    1884;   how  much  did  he  owe,  the 
rate  of  interest  being  8^  ?  Ans.  $228.76. 


Interest.  221 

320.  CASE  III.  —  When  the  time  is  expressed  in  years, 
months,  and  days. 

In  this  case  wo  reduce  the  time  to  months,  thus  : 
1°.  Since  12  months  make  1  year,  we  multiply  the  number  of 
years  by  12,  and  to  the  product  add  the  number  of  months. 

Thus  :  1  yr.  3  mo.  ~  15  mo.  ;  3  yr.  5  mo.  =  41  mo.  ;  etc. 

2°.  Since  30  days  are  reckoned  as  a  month,  3  days  is  jV,  or  .1 
of  a  month  ;  hence,  we  divide  the  number  of  days  by  3,  which  re- 
duces them  to  tenths  of  a  month. 

Thus  :  15  da.=  .5  mo.;  27  da.=  .9  mo.  ;  11  da.==  .3f  mo.  ;  etc. 

In  this  manner  reduce  to  months  : 

1.  2  yr.  3  mo.  6  da.  Ans.  27.2    mo. 

2.  1  yr.  7  mo.  21  da.  Ans.  19.7    mo. 

3.  4  yr.  9  mo.  3  da.  Ans.  57.1    mo. 

4.  5  yr.  2  mo.  19  da.  Ans.  62.6J-  mo. 

5.  1  yr.  1  mo.  1  da.  Ans.  13.  OJ  mo. 

6.  5  yr.  28  da.  Ans.  60.9^-  mo. 

7.  7  mo.  14.  da.  Ans.    7.4  J  mo. 

8.  3  mo.  22  da.  Ans.    3.7-J-  mo. 

9.  26  da.  Ans.      .S-f  mo. 
10.  95  da.  Ans.     3.  If  mo. 

321,  1.  What  is  the  interest  of  $180.60  for  2  yr.  3  mo.  6 
da.  at  6%  ? 

ExpLANATioN.-Since  the  in-  OPERATION. 
terest  of  any  sum  for  one  year  . 

At  6fi  is   -06  of    the   principal, 

$180.60  X  -06  gives  the  interest  .06     Rate 


for  lyr.  or  12  mo.  This  product,  12)^10.8360  Int.  for  1  yr. 

divided  by  12,  gives  the  interest 

for  1  ato.,  which,  multiplied  by  -        -^  Int-  for  1  mo' 

27.2,  the  given  time  expressed  27.!  No.  of  mo. 

in  months,  gives  $24.5610,  the  $24.56160  Required  Int. 
required  interest. 


.06 

27.2 


24.56160 


222  Intermediate  Arithmetic. 

EXPLANATION. — Draw  a  verti- 
cal line,  on  the  right  place  the  2d   OPERATION_ 
principal,  the  rate,  and  number 

of  months,  and  on  the  left,  12.  .180.00  15.05 

Now  cancel  common  factors,  and 
multiply.  lr 

Dividing  180.60  by  12,  we 
get  15.05,  which,  multiplied  by 
.06  and  then  by  27.2,  gives 
$24.5616. 

Hence,  the 

RULE. — Multiply  the  principal  by  the  rate  per  cent,  di- 
vide the  product  by  12,  and  multiply  the  quotient  by  the 
time  expressed  in  months. 

Or,  Draw  a  vertical  line,  on  the  right  place  the  principal, 
rate,  and  number  of  months;  on  the  left,  12  ;  cancel  common 
factors  on  opposite  sides  of  the  line,  and  divide  the  product 
of  the  remaining  factors  on  the  right  by  the  remaining  term, 
if  any,  on  the  left. 

WRITTEN  EXERCISES. 

322.  2.  What  is  the  interest  of  $620  for  1  yr.  8  mo. 
12  da.  at  6^  ?  Ans.  ? 

3.  What  is  the  interest  of  $840.60  for  3  yr.  5  mo.  6 
da.  at  7%?  .   Ans.  $202.0242. 

4.  What  is  the  amount  of  $375  for  2  yr.  6  mo.  21  da. 
at  8^  ?  Ans.      ? 

5.  What  is  the  amount  of  $656.84  for  4  yr.  10  mo. 
15  da.  at  6%  ?  Ans.  $848.9657. 

What  is  the  interest  of: 

6.  $184.80  for  1  yr.  1  mo.   10  da.  at  9^  ?  Ans.  $18.48. 

7.  $321.70  for  4  yr.  3  mo.  27  da.  a£  4%  ?  Ans.  $55.654. 

8.  $208.44  for  7  yr.  8  mo.  15  da.  at  b%  ?  Ans.  $80.336. 


Interest.  223 


9.  $1365.40  for  11  mo.  27  da.  at  b%  ?  Am.  ? 

10.  $200.20  for  2  yr.  24  da.  at  10%  ?  Am.  ? 

11.  $240.60  for  1  yr.  2  mo.  19  da.  at  5%  ?  Ans.  ? 

12.  $360.48  for  5  mo.  17  da.  at  1%  ?  <4n*.  ? 

What  is  the  amount  of: 

13.  $1020.96  for  3  yr.  7  mo.  18  da.  at  5%  ? 

.4ns.  $1206.4344. 

14.  $672.24  for  2  yr.  2  mo.  25  da.  at  6%  ?4?w.  $762.4322. 

15.  $145.20  for  1  yr.  9  mo.  27  da.  at  VL\%  ? 

-4ns.  $178.323+. 

16.  $2500  for  7  mo.  20  da.  at  5%  ?  Ana.  $2579.86. 

17.  $100.25  for  63  da.  at  1%  ?  Ans.  $101.478+. 

18.  What  is  the  interest  of  $1630  from  April  1,  187-S, 
to  Oct.  10,  1882,  at  6%  ?  Ans.  $442.545. 

19.  If  $2150  are  placed  at  interest  May  10,  1877,  what 
amount  will  be  due  Jan.  1,  1881,  at  6fo  ?  Ans.  $2619.775. 

20.  What  is   the  interest    of  $540.36    from   April  25, 
1869,  to  December  15,  1872,  at  6%  ?          Am.  $117.9786. 

21.  What  is  the  amount  of  $360.48  from  Aug.  12,  1869, 
to  March  27,  1872,  at  8%  ?  Am.  $436.1808. 

22.  A  man  borrowed  $480.20  July  13,  1869,  and  kept 
it  until  Feb.  4,  1873;  how  much  did  he  then  owe  at  6%  ? 

Ans.    ? 

23.  How  much  will  an  account  of  $175.50  amount  to 
if  contracted  Nov.  17,  1875,  and  settled  May  5,  1885,  at 
8%  interest?  An*.   ? 

323.  CASE  IV. --When  the  time  is  expressed  in  days. 

EXERCISES. 

1.  What  is  the  interest  of  $450  at  8%  for  91  days? 
We  may  reduce  the  days  to   months  and  proceed  as 


224  Intermediate  Arithmetic. 

in  the  last  case  ;  but  the  following  method  of  solution 
is  generally  preferable : 

EXPLANATION. — Multiplying  $450  by   .08  OPERATION. 

gets  the  int.  for  1  yr.  or  360  da. ;  dividing  A $0     5 


this  product  by  360  gets  the  int.  for  1  da.,       ^^^ 
which,  multiplied  by  91,  gives  the  required 


interest. 


91 


Hence,  the  Ans.  $9.10. 

RULE. — Draw  a  vertical  line,  on  the  right  of  it  place  the 
principal,  rate,  and  days;  on  the  left,  360,  and  proceed  ac- 
cording to  the  rule  of  cancellation. 

What  is  the  interest  of: 

2.  $450  for  63  da.  at  6%  ?  Ans.  $4.725. 

3.  $278.68  for  36  da.  at  10%  ?  Ans.  $2.7868. 

4.  $600  for  93  da.  at  b%  ?  Ans.  $7.75. 

5.  $720  for  125  da.  at  7%  ?  Ans.  $17.50. 

6.  $480.60  for  148  da.  at  9%  ?  Ans.  $17.7822. 

7.  $563.25  for  200  da.  at  12%?  Ans.  $37.55. 

324.  QUESTIONS  FOR  REVIEW. 

What  is :  1.  Interest  ?  2.  The  Principal  ?  3.  The  rate  per  cent  ? 
4.  The  amount  ? 

What  is  the  rule  for  finding  interest  when  the  time  is  expressed 
in  :  1.  Years  ?  2.  Years  and  months  ?  3.  Years,  months,  and 
days?  4.  Days? 


I    ~~J  A    I   — * 

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THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


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